Search Results for Probability

Biographies

1. Ito biography
   
   • It was during his student years that he became attracted to probability theory.
     
   • In [My Sixty Years in Studies of Probability Theory : acceptance speech of the Kyoto Prize in Basic Sciences (1998).',3)">3] he explains how this came about:- .
     
   • Although I knew that probability theory was a means of describing such phenomena, I was not satisfied with contemporary papers or works on probability theory, since they did not clearly define the random variable, the basic element of probability theory.
     
   • At that time, few mathematicians regarded probability theory as an authentic mathematical field, in the same strict sense that they regarded differential and integral calculus.
     
   • When I was a student, there were few researchers in probability; among the few were Kolmogorov of Russia, and Paul Levy of France.
     
   • Accordingly, I was able to continue studying probability theory, by reading Kolmogorov's Basic Concept of Probability Theory and Levy's Theory of Sum of Independent Random Variables.
     
   • At that time, it was commonly believed that Levy's works were extremely difficult, since Levy, a pioneer in the new mathematical field, explained probability theory based on his intuition.
     
   • In 1940 he published On the probability distribution on a compact group on which he collaborated with Yukiyosi Kawada.
     
   • The background to Ito's famous 1942 paper On stochastic processes (Infinitely divisible laws of probability) which he published in the Japanese Journal of Mathematics is given in [Citation for the Kyoto Prize in Basic Sciences awarded to Kiyosi Ito by the Inamori Foundation (1998).',2)">2]:- .
     
   • In 1923, against this scientific background, Wiener defined probability measures in path spaces, and used the concept of Lebesgue integrals to lay the mathematical foundations of stochastic analysis.
     
   • Volume 20 of the Proceedings of the Imperial Academy of Tokyo contains six papers by Ito: (1) On the ergodicity of a certain stationary process; (2) A kinematic theory of turbulence; (3) On the normal stationary process with no hysteresis; (4) A screw line in Hilbert space and its application to the probability theory; (5) Stochastic integral; and (6) On Student's test.
     
   • In the following year he published his famous text Probability theory.
     
   • In this book, Ito develops the theory on a probability space using terms and tools from measure theory.
     
   • Ito gives a wonderful description mathematical beauty in [My Sixty Years in Studies of Probability Theory : acceptance speech of the Kyoto Prize in Basic Sciences (1998).',3)">3] which he then relates to the way in which he and other mathematicians have developed his fundamental ideas:- .
     
   • A recent monograph entitled Ito's Stochastic Calculus and Probability Theory (1996), dedicated to Ito on the occasion of his eightieth birthday, contains papers which deal with recent developments of Ito's ideas:- .
     
   • For almost all modern theories at the forefront of probability and related fields, Ito's analysis is indispensable as an essential instrument, and it will remain so in the future.
     

2. Cantelli biography
   
   • However, it had become less important over the years and was not at the forefront of research at the time Cantelli worked there [Electronic Journal for History of Probability and Statistics 1 (1) (2005).',8)">8]:- .
     
   • Cantelli's work in astronomy involved statistical analysis of data and his interests turned more towards the statistical style of mathematics and to applications of probability to astronomy and other areas.
     
   • In particular he became interested in actuarial and social applications of probability theory.
     
   • In 1903 took a job as an actuary at the Istituti di Previdenza where he undertook research into probability theory publishing some important papers, some which we mention below.
     
   • He founded the Istituto Italiano degli Attuari for the applications of mathematics and probability to economics.
     
   • His later work was all on probability, frequency distributions, actuarial science and applications of probability theory.
     
   • Around 1900 progress in mathematical probability theory was in large part due to the French and to the Russians, and progress in mathematical statistics was mainly due to the British.
     
   • The years between the two World Wars seem to have been a 'golden age' of research on the foundations of probability in Italy.
     
   • from 1900 to 1915 [there was] an interest in the foundational problems of probability, a conviction that logic would play a role in resolving these problems, and varying conceptions of logic, all of which predate and serve as a historical foundation for this 'golden age'.
     
   • For some decades F P Cantelli was one of the most prominent Italian contributors to the mathematical theory of probability.
     
   • Although his name is frequently connected with the name of E Borel, Cantelli's approach to probability is very different from that of Borel.
     
   • Cantelli's first publications on probability examined the foundations of the subject in connection with logic, for example in Sui fondamenti del calcolo delle probabilita (1905).
     
   • He continued to develop these ideas in Sulla probabilita comme limite di frequenza (On probability seen as a limit of frequencies) and Su due applicazioni di un teorema di G Boole alla statistica matematica (On two applications of a Boole's theorem to mathematical statistics) both published in the following year.
     
   • These papers contributed to the debate among some Italian mathematicians about the possibility of defining probability in terms of relative frequencies.
     
   • Regazzini writes [Electronic Journal for History of Probability and Statistics 1 (1) (2005).',8)">8]:- .
     
   • he formulated an abstract theory of probability shortly before the publication of Kolmogorov's 'Grundbegriffe' so ..
     
   • he was not in a position to deal with random variables as measurable functions and, moreover, considered, implicitly, probability as a completely additive function on a family of events, without emphasizing the role of such a hypothesis in restricting the class of the admissible probability assessments.
     

3. Keynes biography
   
   • He worked mostly on his own work, devoting all his spare time to the study of the theory of probability.
     
   • He then submitted a dissertation on probability for a Fellowship at King's College.
     
   • Using the detailed comments on his probability dissertation by both Johnson and Whitehead, Keynes worked hard to improve it.
     
   • After submitting a new version of his dissertation on probability, Keynes was elected to a Fellowship in March 1909.
     
   • Russell, writing about the book which Keynes eventually published on probability, praised the work highly:- .
     
   • Keynes was appointed secretary of a Commission to examine Indian Finance and Currency in 1913 and he began to seek a publisher for his major treatise on probability based on his fellowship dissertation.
     
   • In particular his treatise on probability had to be put to one side until the war was over.
     
   • In 1920 Keynes began to prepare his Treatise on Probability for publication.
     
   • In this work he argues that probability is a logical relation and so it is objective.
     
   • A statement involving probability relations has a truth-value independent of people's opinions.
     
   • In 1926 Ramsey published a paper Truth and probability arguing against these arguments of Keynes.
     
   • 25 (1) (1994), 97-121.',17)">17] examines the two points of view of Keynes and Ramsey on probability.
     
   • Other important ideas discussed by Keynes in Treatise on Probability is that probability relations forms only a partially ordered set in the sense that two probabilities cannot necessarily always be compared.
     
   • Keynes also argues that probability is a basic concept which cannot be reduced to other concepts.
     
   • Keynes: Probability Preface .
     
   • Keynes: Probability Introduction .
     

4. Renyi biography
   
   • In 1952, in addition to his other roles, he was appointed as a professor at the Department of Probability and Statistics of Eotvos Lorand University in Budapest.
     
   • Renyi worked on probability theory which was to be his main research topic throughout his life, but his interests were broad and also covered statistics, information theory, combinatorics, graph theory, number theory and analysis.
     
   • In the hands of writers like Linnik, Erdos and Renyi, the theory of numbers is not clearly distinguished from the theory of probability.
     
   • Thus, when Renyi is referred to as a great applied probabilist, this is partly because of his interests in probability applied to other parts of mathematics.
     
   • Probability theory and its applications had been neglected in the curriculum of Hungarian universities until very recently when the author started to lecture regularly on these topics.
     
   • It is thus the first modern Hungarian text book on probability theory and offers an excellent introduction into this field.
     
   • A French edition appeared in 1966 and an English edition, containing three new sections, was published as Probability theory in 1970.
     
   • Another book, also published in 1970, was Foundations of probability.
     
   • This book developed a totally different approach to probability, based on the concept of conditional probability space, from any other book previously published on the topic.
     
   • In this style he published Dialoge uber Mathematik (1967) and Letters on probability (Hungarian edition 1969, English translation 1972).
     
   • This work is a fascinating semi-popular, semi-historical account of some of the early ideas on probability.
     
   • David Kendall in [Journal of Applied Probability 7 (1970), 509-522.',10)">10] gives us a little more understanding into Renyi's ways of working.
     

5. Laplace biography
   
   • Not only had he made major contributions to difference equations and differential equations but he had examined applications to mathematical astronomy and to the theory of probability, two major topics which he would work on throughout his life.
     
   • established his style, reputation, philosophical position, certain mathematical techniques, and a programme of research in two areas, probability and celestial mechanics, in which he worked mathematically for the rest of his life.
     
   • Laplace served on a committee set up to investigate the largest hospital in Paris and he used his expertise in probability to compare mortality rates at the hospital with those of other hospitals in France and elsewhere.
     
   • In 1795 the Ecole Normale was founded with the aim of training school teachers and Laplace taught courses there including one on probability which he gave in 1795.
     
   • after a general introduction concerning the principles of probability theory, one finds a discussion of a host of applications, including those to games of chance, natural philosophy, the moral sciences, testimony, judicial decisions and mortality.
     
   • the small probability of collision of the Earth and a comet can become very great in adding over a long sequence of centuries.
     
   • The first book studies generating functions and also approximations to various expressions occurring in probability theory.
     
   • The second book contains Laplace's definition of probability, Bayes's rule (so named by Poincare many years later), and remarks on moral and mathematical expectation.
     
   • The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares, Buffon's needle problem, and inverse probability.
     
   • Applications to mortality, life expectancy and the length of marriages are given and finally Laplace looks at moral expectation and probability in legal matters.
     
   • Later editions of the Theorie Analytique des Probabilites also contains supplements which consider applications of probability to: errors in observations; the determination of the masses of Jupiter, Saturn and Uranus; triangulation methods in surveying; and problems of geodesy in particular the determination of the meridian of France.
     

6. Bernoulli Jacob biography
   
   • Jacob Bernoulli's first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687.
     
   • By 1689 he had published important work on infinite series and published his law of large numbers in probability theory.
     
   • The interpretation of probability as relative-frequency says that if an experiment is repeated a large number of times then the relative frequency with which an event occurs equals the probability of the event.
     
   • The work was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability.
     
   • In the book Bernoulli reviewed work of others on probability, in particular work by van Schooten, Leibniz, and Prestet.
     
   • There are interesting thoughts on what probability really is [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .
     
   • probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers ..
     
   • Bernoulli greatly advanced algebra, the infinitesimal calculus, the calculus of variations, mechanics, the theory of series, and the theory of probability.
     

7. Feller biography
   
   • Feller went to Copenhagen where he remained until 1934, then he moved to the University of Stockholm where he joined the probability group headed by Harald Cramer.
     
   • Feller's treatise on probability is one of the great masterpieces of mathematics of all time.
     
   • Feller worked on mathematical probability using Kolmogorov's measure theoretic formulation.
     
   • His approach was pure mathematical but he did study applications of probability, particularly to genetics.
     
   • Thus if it is supposed that the probability that each individual in a time interval of length dt has probability l dt of producing a second individual, the exact value of the probability of having n individuals at time t is found.
     
   • Other papers written by Feller while still at Brown University include: On the time distribution of so-called random events (1940), On the integral equation of renewal theory (1941), On A C Aitken's method of interpolation (1943), The fundamental limit theorems in probability (1945) and Note on the law of large numbers and "fair" games (1945).
     
   • Feller's most important work was Introduction to Probability Theory and its Applications (1950-61), a two volume work which he frequently revised and improved with new approaches, new examples and new applications.
     
   • This fascinating book will remain a standard textbook of mathematical probability for many years to come.
     

8. Linnik biography
   
   • He worked in Leningrad for the rest of his life organising the chair of probability theory there and founding the world famous Leningrad school of probability and mathematical statistics.
     
   • His main research topics were number theory, probability theory and mathematical statistics.
     
   • After his early concentration on number theory, from 1947 onwards Linnik embarked on a deep study of probability.
     
   • From that time on he undertook research in three areas, namely probability, mathematical statistics and the analytic theory of numbers.
     
   • In 1950 he introduced the concepts of probability into number theory and introduced the dispersion method in number theory.
     
   • Later Linnik made major contributions to probability with his work on limit theorems and was the first to use powerful techniques from analysis in mathematical statistics.
     
   • He wrote many papers on the decomposition of probability laws and again collected his results into a monograph Decompositions of probability laws first published in Russian in 1960.
     
   • A volume has also been published of his work on Probability theory (1981) and on Mathematical statistics (1982).
     

9. Mises biography
   
   • His Institute rapidly became a centre for research into areas such as probability, statistics, numerical solutions of differential equations, elasticity and aerodynamics.
     
   • Von Mises worked on fluid mechanics, aerodynamics, aeronautics, statistics and probability theory.
     
   • He classified his own work, not long before his death, into eight areas: practical analysis, integral and differential equations, mechanics, hydrodynamics and aerodynamics, constructive geometry, probability calculus, statistics and philosophy.
     
   • His most famous, and at the same time most controversial, work was in probability theory.
     
   • One has even forgiven him his theory of probability.
     
   • that the basis for the applicability of the results of the mathematical theory of probability to real 'random phenomena' must depend on some form of the frequency concept of probability, the unavoidable nature of which has been established by von Mises in a spirited manner.
     
   • von Mises' notion of a random sequence in the context of his approach to probability theory.
     
   • [The author claims] that the acceptance of Kolmogorov's rival axiomatisation was due to a different intuition about probability getting the upper hand, as illustrated by the notion of a martingale.
     

10. Reichenbach biography
    
    • In 1915 he received his doctorate from the University of Erlangen for his thesis on philosophical aspects of the theory of probability Der Begriff der Wahrscheinlichkeit fur die mathematische Darstellung der Wirklichkeit.
      
    • Basically, two tasks face any treatment of the foundations of probability: (I) the establishment of laws of consistency of probability statements (laws permitting the derivation of new probabilities from given probabilities); and (II) the formulation of explicit rules for assigning probabilities in the first place, in situations where no probability is given.
      
    • Both (I) and (II) entail the question of the meaning of probability and the problem of its application.
      
    • As the author shows, the latter problem has a unique form in probability: the application problem in other sciences makes use of probability.
      
    • Reichenbach attempted to define probability as the limit of a frequency but many criticised this approach.
      
    • We have seen that Reichenbach wrote on induction, probability and the philosophy of science.
      

11. Doob biography
    
    • Hotelling managed to obtain a Carnegie fellowship to enable Doob to remain at Columbia University and work with him on probability during the year 1934-35.
      
    • Doob's work was in probability and measure theory, in particular he studied the relations between probability and potential theory.
      
    • Monthly 105 (1) (1998), 28-35.',1)">1] looks at many of the areas of probability to which Doob made major contributions such as separability, stochastic processes, martingales, optimal stopping, potential theory, and classical potential theory and its probabilistic counterpart.
      
    • probability theory is simply a branch of measure theory, with its own special emphasis and field of application ..
      
    • His interest in potential theory went back to 1955 when he was invited to speak at the Berkeley Symposium on Probability and Statistics.
      
    • He asked Doob to cooperate with him in writing the sections on probability theory but in the end Doob wrote the whole book.
      
    • Graduate students and researchers in probability or classical analysis will find much to learn from this fine book by a master of both areas.
      
    • his fundamental work in establishing probability as a branch of mathematics and for his continuing profound influence on its development.
      

12. Kolmogorov biography
    
    • Another milestone occurred in 1925, namely Kolmogorov's first paper on probability appeared.
      
    • His monograph on probability theory Grundbegriffe der Wahrscheinlichkeitsrechnung published in 1933 built up probability theory in a rigorous way from fundamental axioms in a way comparable with Euclid's treatment of geometry.
      
    • After mentioning the highly significant paper Analytic methods in probability theory which Kolmogorov published in 1938 laying the foundations of the theory of Markov random processes, they continue to describe:- .
      
    • The Department of Probability and Statistics was set up at the Institute and Kolmogorov was appointed as Head of Department.
      
    • He thus demonstrated the vital role of probability theory in physics.
      
    • 22 (1) (1990), 31-100.',10)">10] notes Kolmogorov's major part in setting up the theory to answer the probability part of Hilbert's Sixth Problem "to treat ..
      
    • by means of axioms those physical sciences in which mathematics plays an important part; in the first rank are the theory of probability and mechanics" in his 1933 monograph Grundbegriffe der Wahrscheinlichkeitsrechnung.
      

13. Markov biography
    
    • After 1900 Markov applied the method of continued fractions, pioneered by his teacher Pafnuty Chebyshev, to probability theory [The St Petersburg school of number theory (American Mathematical Socity, Providence, RI, 2005).',4)">4]:- .
      
    • Markov was the most elegant spokesman for Chebyshev's ideas and directions of research in probability theory.
      
    • Especially remarkable is his research relating to the theorem of Jacob Bernoulli known as the Law of Large Numbers, to two fundamental theorems of probability theory due to Chebyshev, and to the method of least squares.
      
    • He also studied sequences of mutually dependent variables, hoping to establish the limiting laws of probability in their most general form.
      
    • This work founded a completely new branch of probability theory and launched the theory of stochastic processes.
      
    • A A Markov's classic course on the computation of probabilities, and his original memoirs, models of accuracy and clarity of exposition, contributed to a very large extent to the transformation of the theory of probability into one of the most perfected areas of mathematics, and to the wide dissemination of Chebyshev's methods and directions of research.
      
    • His profound analysis in the spirit of Chebyshev of the dependencies among observed random phenomena allowed Markov to extend probability theory in an essential way through the introduction and investigation of dependent random quantities.
      
    • Although by 1921 he was in such a bad way that he was hardly able to stand, yet he continued to lecture on probability at the university.
      

14. De Finetti biography
    
    • The reference is to his famous subjective theory of probability, which he developed during his most prolific period, that is the one from 1926 to 1931.
      
    • Probability does not exist .
      
    • which conveys his idea that probability is an expression of the observer's view of the world and as such it has no existence of its own.
      
    • Although the idea of probability as a measure of the observer's belief that an event will happen had already been conceived by F P Ramsey in 1926, Bruno de Finetti was unaware of Ramsey's work and, moreover, his chief interest was for coherent probability assessments and not for rational decisions; see the obituary by L Daboni [Bollettino dell\'Unione Matematica Italiana, Serie VII 1-A (1987), 283-308.',4)">4] for more information.
      
    • A "summa" of Bruno de Finetti's revolutionary ideas on probability can be found in the two volumes of his best known book Teoria della Probabilita (1970) which was translated into English in 1975.
      
    • However, his contributions to probability and statistics do not reduce to his subjective approach and in fact they include important results on finitely additive measures, processes with independent increments, sequences of exchangeable variables and associative means; see the review by M D Cifarelli and E Regazzini [Statistical Science 11 (1996), 253-282.',3)">3] for details on these.
      
    • Moreover, Bruno de Finetti had interests and made contributions well outside the field of probability and statistics.
      

15. Cramer Harald biography
    
    • It was not only through his work on number theory that Cramer was led towards probability theory.
      
    • This led him to study probability and statistics which then became the main area of his research.
      
    • In 1927 he published an elementary text in Swedish Probability theory and some of its applications.
      
    • Cramer became interested in the rigorous mathematical formulation of probability in work of the French and Russian mathematicians such as Paul Levy, Sergei Bernstein, and Aleksandr Khinchin in the early 1930s, but in particular the axiomatic approach of Kolmogorov.
      
    • The results of his studies were written up in his Cambridge publication Random variables and probability distributions which appeared in 1937.
      
    • In this classic of statistical mathematical theory, Harald Cramer joins the two major lines of development in the field: while British and American statisticians were developing the science of statistical inference, French and Russian probabilists transformed the classical calculus of probability into a rigorous and pure mathematical theory.
      
    • One finds treated such fields as number theory, function theory, mathematical statistics, probability and stochastic processes, demography, insurance risk theory, functional analysis and the history of mathematics.
      

16. Buffon biography
    
    • He corresponded with Gabriel Cramer on mechanics, geometry, probability, number theory and the differential and integral calculus.
      
    • He next published Memoire sur le jeu de franc-carreau which introduced differential and integral calculus into probability theory.
      
    • The wide range of topics which Buffon wrote on include mathematics, the theory of probability, astronomy and physics, especially optics.
      
    • His most notable contribution to mathematics was a probability experiment which he carried out calculating by throwing sticks over his shoulder onto a tiled floor and counting the number of times the sticks fell across the lines between the tiles.
      
    • This experiment caused much discussion among mathematicians which helped towards an understanding of probability.
      
    • The needle experiment, described in 1777, was not the only problem in probability that Buffon examined.
      
    • Also in 1777 he attempted to calculate the probability that the sun would continue to rise after having been observed to rise n days in a row; see [Arch.
      

17. Lambert biography
    
    • Lambert is also important for his study of the trigonometry of triangles on surfaces, his work on perspective and cartography, as well as his contributions to the theory of probability.
      
    • His contributions to probability are evaluated by Garibaldi and Penco in [Rend.
      
    • In his own fundamental philosophical opus, the "Neues Organon", Lambert developed a noteworthy theory of logical probability that, to our knowledge, has thus far escaped the attention of eminent scholars in the field such as Keynes.
      
    • Logical probability is a 'third general type of probability' that follows in order of exposition the 'a priori probability' typical of games of change and the 'a posteriori probability' of statistics.
      

18. Polya biography
    
    • In the following year he returned to Budapest where he was awarded a doctorate in mathematics having studied, essentially without supervision, a problem in the theory of geometric probability.
      
    • His basic research contributions span complex analysis, mathematical physics, probability theory, geometry, and combinatorics.
      
    • The following year, in addition to papers on these topics, he published on astronomy and probability.
      
    • In probability Polya looked at the Fourier transform of a probability measure, showing in 1923 that it was a characteristic function.
      
    • He considered a d-dimensional array of lattice points where a point moves to any of its neighbours with equal probability.
      
    • He asked whether given an arbitrary point A in the lattice, a point executing a random walk starting from the origin would reach A with probability 1.
      

19. Chebyshev biography
    
    • Brashman was particularly interested in mechanics but his interests were wide ranging and, in addition to courses on mechanical engineering and hydraulics, he taught his students the theory of integration of algebraic functions and the calculus of probability.
      
    • The thesis was on the theory of probability, and in it he developed the main results of the theory in a rigorous but elementary way.
      
    • Laplace had found and studied the Hermite polynomials in the course of his discoveries in probability theory during the early nineteenth century.
      
    • His work arose out of the theory of least squares approximation and probability; he applied his results to interpolation, approximate quadrature and other areas.
      
    • We have mentioned some contributions that Chebyshev made to the theory of probability.
      
    • Twenty years later Chebyshev published On two theorems concerning probability which gives the basis for applying the theory of probability to statistical data, generalising the central limit theorem of de Moivre and Laplace.
      

20. Gnedenko biography
    
    • It was during this period that Gnedenko published his first papers on probability and statistics.
      
    • He became deeply interested in probability theory after attending seminars by Kolmogorov and Khinchin.
      
    • He held these posts until 1960 when he returned to Moscow University, becoming Head of the Department of Probability Theory in 1966.
      
    • One of Gnedenko's most famous books is Course in the Theory of Probability which first appeared in 1950.
      
    • Written in a clear and concise manner, the book was very successful in providing a first introduction to probability and statistics.
      
    • The book is written on a mature level, but little probability theory beyond very elementary concepts or very intuitive ones is presupposed.
      
    • In his early work Gnedenko had been interested in probability as an abstract topic.
      

21. Borel biography
    
    • Also in 1921 he was appointed to succeed Joseph Boussinesq in the chair of Probability and Mathematical Physics and, in the following year, he founded the Institut de Statistique de l'Universite de Paris (Institute of Statistics at the University of Paris).
      
    • After 1905 he became interested in the theory of probability.
      
    • In addition to many textbooks, Borel published more than fifty papers between 1905 and 1950 on the calculus of probability.
      
    • He stressed the important and practical value of probability theory.
      
    • He preferred to elucidate these applications instead of looking for an axiomatization of probability theory.
      
    • Nevertheless, he was interested in a clarification of the probability concept.
      

22. Bertrand biography
    
    • Bertrand published many works on differential geometry and on probability theory.
      
    • He wrote a number of notes on the theory of probability and on the reduction of data from observations.
      
    • He published these notes starting around 1875 and, after a short break of three years from 1884, he began publishing further notes on probability.
      
    • It concerns the probability that an arbitrary chord of a circle is longer than a side of an equilateral triangle inscribed in the circle.
      
    • General comments on this work and Bertrand's other notes on probability are made in [La France Mathematique.
      
    • Introduction to Bertrand's work on probability .
      

23. Pitt biography
    
    • Coastal Command he used probability theory, and the newly developing operational research, to devise methods for attacking German U-boats.
      
    • However he took the opportunity to write two classic texts publishing Tauberian theorems in 1958 and Integration, measure and probability in 1963.
      
    • As the author states in the preface, the purpose of this book is to provide an introduction to the modern theory of probability and the fundamental ideas and techniques on which it is based, namely, those of measure and integration.
      
    • The book Integration, measure and probability appeared after he had spent the year 1962-63 as a visiting professor at Yale University.
      
    • This compact account develops the theory as it applies to abstract spaces, describes its importance to differential and integral calculus, and shows how the theory can be applied to geometry, harmonic analysis, and probability.
      
    • The second part also has three chapters: Each one of them considers an area of application: geometry, harmonic analysis and probability.
      

24. Bachelier biography
    
    • One of his courses was Probability calculus with applications to financial operations and analogies with certain questions from physics.
      
    • In this course he may have drawn out the similarities between the diffusion of probability (the total probability of one being conserved) and the diffusion equation of Fourier (the total heat-energy being conserved).
      
    • It seems extraordinary that Levy was, apparently, unfamiliar with Bachelier's work as Bachelier had by this time (1926) published 3 books and some 13 papers on probability and regarded showing how a continuous distribution could be derived from a discrete distribution as his most important achievement.
      
    • measure theory and axiomatic probability) although, his results were basically correct.
      
    • Bachelier's work is remarkable for herein lie the theory of Brownian Motion (one of the most important mathematical discoveries of the 20th century), the connection between random walks and diffusion, diffusion of probability, curves lacking tangents (non-differentiable functions), the distribution of the Wiener process and of the maximum value attained in a given time by a Wiener process, the reflection principle, the pricing of options including barrier options, the Chapman-Kolmogorov equations in the continuous case, .
      

25. Dynkin biography
    
    • His work at this time was partly in algebra and partly in probability.
      
    • For ten years he worked both on the theory of Lie algebras and on probability theory although his main work during this period was in algebra.
      
    • In 1945 he solved a problem on Markov chains suggested by Kolmogorov and his first publication in probability resulted.
      
    • and he became an assistant professor of Kolmogorov's who held the Probability Chair.
      
    • From the time he was appointed to the chair, Dynkin's work became more and more devoted to probability theory.
      
    • His work from this period is contained in two major books Foundations of the Theory of Markov Processes (1959) and Markov Processes (1963) which have become classics of probability theory.
      
    • Eugene B Dynkin has made major contributions to the theory of Lie algebras and to probability theory.
      
    • Even though Dynkin has dealt with quite concrete probability problems, one of his strengths is his ability to build general theories and an apparatus to answer broad questions ..
      

26. Suetuna biography
    
    • One purpose of his visit had been to learn probability and statistics, for these topics did not have active researchers in Japan at this time and Tokyo University had sent him to Germany to gain expertise in these areas.
      
    • Suetuna did not spent a great deal of time attending lectures on probability and statistics, preferring to work on his own research topics.
      
    • However he did read books on probability and statistics while in Germany and by the time he returned to Japan he had gained considerable expertise in probability and statistics despite concentrating on his research topics in algebra and number theory.
      
    • He continued to publish research papers on topics related to Artin's 1927 paper and he also wrote several books: one on algebra and number theory, one on analytic number theory, and one on probability.
      
    • All were based on lecture courses he gave, the probability book being based on a ten lecture course he was invited to give at Hokkaidu University in 1940.
      

27. Lukacs biography
    
    • Wald was working on statistics and probability and he persuaded Lukacs to take an interest in this topic too.
      
    • The two worked on probability and wrote a number of joint papers.
      
    • Applied Probability 25 (1988), 641-646.
      
    • Applied Probability 25 (1988), 641-646.
      
    • We shall miss Eugene greatly, not only for his contributions to probability and statistics but also as a colleague, a friend and as a human being of integrity.
      
    • Jointly with Z W Birnbaum, he was the founding editor of the Academic Press Series in Probability and Mathematical Statistics (1962-85).
      

28. Levy Paul biography
    
    • Probability 1 (1) (1971), 1-18.',9)">9]):- .
      
    • At that time there was no mathematical theory of probability - only a collection of small computational problems.
      
    • If there is one person who has influenced the establishment and growth of probability theory more than any other, that person must be Paul Levy.
      
    • Probability 1 (1) (1971), 1-18.',9)">9], gives a very colourful description of Levy's contributions:- .
      
    • Not only did Levy contribute to probability and functional analysis but he also worked on partial differential equations and series.
      
    • Probability 1 (1) (1971), 1-18.',9)">9] in these words:- .
      

29. De Moivre biography
    
    • De Moivre pioneered the development of analytic geometry and the theory of probability.
      
    • In fact it was Francis Robartes, who later became the Earl of Radnor, who suggested to de Moivre that he present a broader picture of the principles of probability theory than those which had been presented by Montmort in Essay d'analyse sur les jeux de hazard (1708).
      
    • Any number of letters a, b, c, d, e, f, etc., all of them different, being taken promiscuously as it happens: to find the probability that some of them shall be found in their places according to the rank they obtain in the alphabet; and that others of them shall at the same time be displaced.
      
    • In fact in A history of the mathematical theory of probability (London, 1865), Todhunter says that probability:- .
      
    • the first occurrence of the normal probability integral.
      

30. Mazurkiewicz biography
    
    • His main work was in topology and the theory of probability.
      
    • Throughout his career Mazurkiewicz was interested in the theory of probability.
      
    • He also considered axiom systems for probability theory, publishing different versions in the years 1933 and 1934.
      
    • It was during this period of occupation that Mazurkiewicz wrote a treatise on probability including his own results on the topic.
      
    • Mazurkiewicz escaped with his life, but the manuscript of his treatise on probability was destroyed as the buildings burned.
      
    • However, he attempted to rewrite his treatise on probability which had been burnt in the destruction.
      

31. Schmetterer biography
    
    • Schmetterer's interests turned towards probability.
      
    • In 1948 I was asked by the mathematicians in Vienna to give lectures in probability theory ..
      
    • He had lectured in the field of probability from a standpoint of the theory of errors.
      
    • I did not know anything about probability at this time.
      
    • It begins with an introductory chapter on Mathematical Probability, well and clearly written though, as the author points out, not designed to give a completely rigorous mathematical treatment.
      
    • The famous algebraists in this group were a major influence on Schmetterer who attended lectures by Artin and started research on probability on algebraic structures.
      

32. Bayes biography
    
    • Bayes set out his theory of probability in Essay towards solving a problem in the doctrine of chances published in the Philosophical Transactions of the Royal Society of London in 1764.
      
    • In an introduction which he has writ to this Essay, he says, that his design at first in thinking on the subject of it was, to find out a method by which we might judge concerning the probability that an event has to happen, in given circumstances, upon supposition that we know nothing concerning it but that, under the same circumstances, it has happened a certain number of times, and failed a certain other number of times.
      
    • What may the reader expect to find in this Essay? As regards probability, he will expect, of course, some or other version of what has become known as 'Bayes's theorem': and such expectation will indeed be met.
      
    • the first occurrence of a probability logic result involving conditional probability.
      
    • This notebook contains a considerable amount of mathematical work, including discussions of probability, trigonometry, geometry, solution of equations, series, and differential calculus.
      

33. Hajek biography
    
    • He was among the pioneers of unequal probability sampling.
      
    • It was a productive period during which he wrote 20 papers and two books: The theory of Probability Sampling with Applications to Sample Surveys and Probability in Science and Engineering which he wrote jointly with V Dupac.
      
    • Hajek developed the property of sequences of pairs of probability measures from ideas due to de la Vallee Poussin.
      
    • This led to an association with the Charles University of Prague where he was appointed to the Chair of Probability and Statistics in 1964 [Jaroslav Hajek : Collected works of Jaroslav Hajek - with commentary (Chichester, 1998).
      

34. Khinchin biography
    
    • Around 1922 Khinchin took up new mathematical interests when he began to study the theory of numbers and probability theory.
      
    • In 1927 Khinchin was appointed as a professor at Moscow University and, in the same year, he published Basic laws of probability theory.
      
    • Khinchin left Moscow in 1935 to spend two years at Saratov University but returned to Moscow University in 1937 to continue his role of building the school of probability theory there in partnership with Kolmogorov and others, including in particular their student Gnedenko.
      
    • It consists of English translations of two articles: The entropy concept in probability theory and On the basic theorems of information theory which were both published earlier in Russian.
      
    • II (Berkeley, 1961), 1-15.',6)">6] Gnedenko, who was a student of Khinchin, lists 151 publications by Khinchin on the mathematical theory of probability (the list is given again in [Ann.
      

35. Quetelet biography
    
    • He learnt astronomy from Arago and Bouvard and the theory of probability under Joseph Fourier and Pierre Laplace.
      
    • He gave his first course on probability in the academic year 1824-25.
      
    • He also began to give public lectures at the Museum in Brussels on topics such as geometry, probability, physics, and astronomy.
      
    • Since absolute certainty is impossible, and we can speak only of the probability of the fulfilment of a scientific expectation, a study of this theory should be a part of very man's education.
      
    • This probability may be considered as giving, in cities, the measure of the apparent tendency to marriage of a Belgian aged 25 to 30.
      

36. Doeblin biography
    
    • Wolfang Doeblin is arguably one of the four major contributors to probability theory in the first half of the 20th century up to World War II (the other three are Khinchin, Kolmogorov and Levy).
      
    • The book by Doob [Stochastic Processes (New York, 1953).',3)">3] from 1953 has been crucial for the development of probability theory; for a large part of its contents on Markov chains and processes, Doeblin's work is the base.
      
    • In Doeblin's mine of ideas, the coupling method was paid attention to by very few until the early 1970s; then the time was ripe to explore it, and the method is now a major tool in probability theory, with applications ranging from elementary theory to front research.
      
    • On Doeblin's work concerning sums of independent random variables, Feller [An Introduction to Probability Theory and Its Applications Vol.
      
    • Humaines 119 (1992), 5-51.',6)">6], [H Cohn (ed.), Doeblin and Modern Probability (Providence, RI, 1993).',7)">7], [Ann.
      

37. Jeffreys biography
    
    • We began by attending a lecture course given by Professor Harold Jeffreys on 'Probability'.
      
    • in recognition of his distinguished work in many branches of geophysics, and also in the theory of probability and astronomy.
      
    • In addition to Methods of Mathematical Physics other contributions of his to pure mathematics are contained in Theory of Probability (1939).
      
    • His work in probability is developed along Bayesian lines and again aimed at application in the physical sciences.
      
    • Harold Jeffreys on Probability .
      

38. Bauer biography
    
    • The second part of the book is devoted to probability theory.
      
    • Generally speaking, only probability theory as it pertains to product measure spaces is discussed.
      
    • An English version Probability Theory and Elements of Measure Theory was published in 1972.
      
    • Because of the great popularity the book enjoyed, an extensive reworking and expansion of the sections on probability appeared in English translation as Probability theory in 1996, with the same treatment was given to the sections of measure theory, published in English translation as Measure and integration theory in 2001.
      

39. Castelnuovo biography
    
    • Later in his career at Rome he taught a course on algebraic functions and abelian integrals in which he treated the theory of Riemann surfaces,and courses on non-euclidean geometry, differential geometry, interpolation and approximation, and probability theory.
      
    • He explained why he found probability an interesting topic to teach:- .
      
    • Probability is a science of recent formation; hence in it, better than in other branches of mathematics, one can see the relationship between the empirical contribution and the one given by reasoning, and between the process of inductive and deductive logic used in it.
      
    • Castelnuovo also wrote a book on probability, publishing Calcolo della probabilita in 1919 and a text on the theory of relativity in 1923.
      
    • A summary of his important probability text is given in [Simposio Internazionale de Geometria Algebrica, Rome, 1965 (Rome, 1967), xxxvi-liii.',14)">14].
      

40. Arbuthnot biography
    
    • He translated Huygens' tract De ratiociniis in ludo aleae on probability and extended it by adding to it a few further games of chance such as backgammon, the Royal Oak lottery, raffling, whist, and games with dice.
      
    • It was the first work on probability published in English and refers to:- .
      
    • the calculation of the quantity of probability.
      
    • This appears to be the first time the word "probability" appears in print.
      
    • This paper, published in the Philosophical Transactions, is perhaps the first application of probability to social statistics and includes the first formal test of significance.
      

41. Johnson biography
    
    • During these nineteen years of holding temporary positions he published three papers on Boolean logic and one on probability.
      
    • Logic is in three volumes, the fourth on probability was never finished, but the parts which were written were published in Mind after his death.
      
    • Johnson viewed probability as expressing logical relations between evidence propositions and hypothesis propositions.
      
    • He was opposed to the frequency interpretation of probability.
      
    • His views on the foundations of probability theory influenced Keynes and others.
      

42. Jevons biography
    
    • An important influence on Jevons while he was studying in London was De Morgan, not in terms of Jevons thoughts on economics but certainly in terms of his thoughts on logic and probability.
      
    • Grattan-Guinness [Studies in the History of Statistics and Probability II (London, 1977), 180-212.',10)">10] suggests that the main difference between their approach was that, although both believed they were studying the laws of thought, Boole had a more algebraic concept of logic while Jevons argued that mathematics proceeds from logic.
      
    • This work made important contributions to probability as well as to logic.
      
    • Although the work contains much in the way of innovative ideas perhaps its weaknesses are illustrated by one of his examples on the use of probability.
      
    • He then naively claims that the probability that the next element discovered will be a metal is (50 + 1)/(64 + 2) = 17/22.
      

43. Coolidge biography
    
    • He also wrote on probability with An Introduction to Mathematical Probability (1925).
      
    • An Introduction to Mathematical Probability is [American National Biography 4 (Oxford, 1999), 424-425.',4)">4]:- .
      
    • it defined probability ..
      

44. David biography
    
    • David published Tables of the ordinates and probability integral of the distribution of the correlation coefficient in small samples in 1938.
      
    • In 1949 David published Probability Theory for Statistical Methods, an elementary textbook in the theory of probability for students of statistics.
      
    • It presents a history of probability and statistical ideas and was republished by Dover Publications in 1998.
      

45. Ramsey biography
    
    • Ramsey made a systematic attempt to base the mathematical theory of probability on the notion of partial belief.
      
    • This work on probability, and also important work on economics, came about mainly because Ramsey was a close friend of Keynes.
      
    • Being a friend of Keynes certainly did not stop Ramsey attacking Keynes' work, however, and in Truth and probability , which Ramsey published in 1926, he argues against Keynes' ideas of an a priori inductive logic.
      
    • Ramsey, proposing a probability measure based on strength of belief, [Routledge Encyclopedia of Philosophy 8 (London, New York, 1998), 44-49.',11)">11]:- .
      

46. Fermat biography
    
    • Fermat's correspondence with the Paris mathematicians restarted in 1654 when Blaise Pascal, Etienne Pascal's son, wrote to him to ask for confirmation about his ideas on probability.
      
    • Their short correspondence set up the theory of probability and from this they are now regarded as joint founders of the subject.
      
    • Fermat however, feeling his isolation and still wanting to adopt his old style of challenging mathematicians, tried to change the topic from probability to number theory.
      
    • This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory.
      

47. Bonferroni biography
    
    • His articles are more of a contribution to probability theory than to simultaneous statistical inference, and the reader in search of a convenient reference for such use might prefer [Simultaneous statistical inference, (New York, 1966).',8)">8].
      
    • Apart from these he also had interests in the foundations of probability.
      
    • He developed a strongly frequentist view of probability denying that subjectivist views can even be the subject of mathematical probability.
      

48. Kingman biography
    
    • When I told him of my interest in probability and statistics, he arranged for me to be taught by Dennis Lindley, perhaps the best teacher of the subject for a generation.
      
    • Two important books by Kingman were published in that year, namely Introduction to Measure and Probability (written jointly with S J Taylor) and The Algebra of Queues.
      
    • Kingman gave a beautiful description of the development of the subject in his 1973 paper Subadditive ergodic theory published in the Annals of Probability.
      
    • I was appointed Professor of Mathematics to raise the profile of probability theory (but not statistics) in the Faculty of Mathematics.
      

49. Crofton biography
    
    • He is perhaps best known, however, for his contributions to geometric probability, integral geometry and Crofton's formula (see for example [Archimede 35 (3) (1983), 110-126.',9)">9] and [Arch.
      
    • Crofton wrote On the theory of local probability in 1868 which he discussed Buffon's needle.
      
    • Perhaps his most famous work on probability, however, was the article that he wrote on that topic for the ninth edition of Encyclopaedia Britannica.
      
    • Crofton's lengthy article Probability which appeared in 1885 is still worth reading and is one of many outstanding articles in what many consider to be the greatest encyclopaedia ever produced.
      

50. Erlang biography
    
    • His interests turned towards the theory of probability and he kept up his mathematical interests by joining the Mathematical Association.
      
    • In 1908 Erlang joined the Copenhagen Telephone Company and began applying probability to various problems arising in the context of telephone calls.
      
    • He published his first paper on these problems The theory of probability and telephone conversations in 1909.
      
    • In addition to his work on probability Erlang was also interested in mathematical tables.
      

51. Schramm biography
    
    • His work in a spectacular series of papers has led to major progress in probability theory, in the theory of percolation and of random walks, as well as in related topics of conformal field theory.
      
    • work in combining analytic power with geometric insight in the field of random walks, percolation, and probability theory in general, especially for formulating stochastic Loewner evolution.
      
    • His research in probability was sparked by his interest in the conjecture that the limit of two-dimensional critical percolation was conformally invariant.
      
    • In trying to understand this limit as well as limits of other models such as the loop-erased walk, Schramm combined classical results in complex variables of C Loewner with probability theory to invent the process now called the Schramm-Loewner evolution (SLE).
      

52. Gateaux biography
    
    • Morale (1924).',13)">13]), Levy takes stock of this definition of integral as mean value as a natural definition of the uniform probability in an infinite set.
      
    • In [Lecons d\'Analyse fonctionnelle (Gauthier-Villars, 1922).',12)">12], Levy has finally admitted that the right formulation for these problems is in a probabilistic framework, and it is impressive to see how in the book (and in particular in Chapter VI), Levy makes use of probability theory to justify the passages to the limit by means of the law of large numbers.
      
    • Interestingly, probability theory does not appear in the notes Levy had published just after the war about his work on the function of lines.
      
    • It is only when he wrote his book [Lecons d\'Analyse fonctionnelle (Gauthier-Villars, 1922).',12)">12] that he understood this natural framework, at the precise moment when he began to be interested in probability theory because he needed to teach it.
      

53. Craig biography
    
    • In 1699 he published Theologiae Christianae Principia Mathematica which applies probability to show that the evidence of the truth of the gospels is diminished through time.
      
    • a formula tantamount to a logistic model for posterior odds: that is, Craig's probability should be understood as the logarithm of the ratio of the probability of the historical testimony as received at the present time, given the historical hypothesis in question, to the probability of the same testimony, given the negation of that hypothesis.
      

54. Neyman biography
    
    • In the academic year 1915-16 Aleksandr Bernstein lectured to him on probability; he strongly influenced Neyman and encouraged him to read Karl Pearson's The Grammar of Science.
      
    • He received a doctorate in 1924 for a thesis on application of probability to agricultural experimentation after being examined by a panel which included Sierpinski and Mazurkiewicz.
      
    • Shortly after his return he had a major paper Outline of a theory of statistical estimation based on the classical theory of probability accepted for publication by the Royal Society.
      
    • There he taught probability and statistics in the mathematics department but aimed to set up a centre to train American statisticians.
      

55. Bernstein Sergi biography
    
    • Some of Bernstein's most important work was in the theory of probability.
      
    • He attempted an axiomatisation of probability theory in 1917.
      
    • Bernstein also studied applications of probability, in particular to genetics.
      

56. Condorcet biography
    
    • His most important work was on probability and the philosophy of mathematics.
      
    • His most important treatise was Essay on the Application of Analysis to the Probability of Majority Decisions (1785).
      
    • This is an extremely important work in the development of the theory of probability.
      

57. Vinogradov biography
    
    • Two of his teachers there, A A Markov and Ya V Uspenskii, both had interests in probability and number theory and Vinogradov's interest in number theory stems from this period.
      
    • He was promoted to professor at the university in 1925, becoming head of the probability and number theory section.
      
    • These were: fundamental questions of analysis and mathematical physics; special areas of function theory of real variables; number theory and Galois theory; probability theory; theoretical mechanics; applied methods of analysis.
      

58. Bunyakovsky biography
    
    • He also brought French probabilistic ideas which formed a basis for the development of probability in the Empire prior to the work of Chebyshev; details are given in [Arch.
      
    • His 1846 book on probability Foundations of the mathematical theory of probability is usually recognised as providing the development of Russian probabilistic terminology.
      

59. Steinhaus biography
    
    • by Steinhaus and Banach, concentrated mainly on functional analysis and its diverse applications, the general theory of orthogonal series, and probability theory.
      
    • As we have noted above, other contributions by Steinhaus were on orthogonal series, probability theory, real functions and their applications.
      
    • In particular he is associated with the theory of independent functions, arising from his work in probability theory, and he was the first to make precise the concepts of "independent" and "uniformly distributed".
      

60. Kac biography
    
    • Kac has pioneered the modern development of mathematical probability, in particular its applications to statistical physics.
      
    • He published a classic text Statistical Independence in Probability, Analysis and Number Theory in 1959.
      
    • To Mark Kac for his important contributions to statistical mechanics and to probability theory and its applications.
      

61. Gosset biography
    
    • Writing in [Studies in the History of Statistics and Probability (London, 1970), 355-404.',8)">8], McMullen says:- .
      
    • In 1934 Gosset had a motor accident, described in [Studies in the History of Statistics and Probability (London, 1970), 355-404.',8)">8]:- .
      
    • McMullen, who was a personal friend, describes Gosset in [Studies in the History of Statistics and Probability (London, 1970), 355-404.',8)">8] as follows:- .
      

62. Burnside biography
    
    • In fact in the latter years of his life he turned to probability theory and his first paper on the subject appeared in 1918.
      
    • He left a complete manuscript of a book on probability which was published as The Theory of Probability in the year after his death.
      

63. Savage biography
    
    • The other main direction of his work was to study gambling as a source to stimulate problems in probability and decision theory.
      
    • The book considers subjective probability and utility.
      
    • It starts with six axioms, which are both motivated and discussed, and from these are deduced the existence of a subjective probability and a utility function.
      

64. Bernoulli Daniel biography
    
    • The first part described the game of faro and is of little importance other than showing that Daniel was learning about probability at this time.
      
    • A second important work which Daniel produced while in St Petersburg was one on probability and political economy.
      
    • By 1731 he was applying for posts in Basel but probability seemed to work against him and he would lose out in the ballot for the post.
      

65. Seidel biography
    
    • He also introduced the concept of nonuniform convergence and applied probability to astronomy.
      
    • An interesting aspect of Seidel's astronomical work involved, as we mentioned above, the use of probability theory.
      
    • He lectured on probability theory, and also on the method of least squares.
      

66. Frechet biography
    
    • It was after going to Strasbourg that he began to become interested in statistics but he only published a small number of articles on probability at this stage, most of his papers being on general analysis and topology.
      
    • However, he taught courses on probability, statistics, and insurance mathematics at Strasbourg.
      
    • Frechet also made important contributions to statistics, probability and calculus.
      

67. Poisson biography
    
    • In Recherches sur la probabilite des jugements en matiere criminelle et matiere civile, an important work on probability published in 1837, the Poisson distribution first appears.
      
    • The Poisson distribution describes the probability that a random event will occur in a time or space interval under the conditions that the probability of the event occurring is very small, but the number of trials is very large so that the event actually occurs a few times.
      

68. Gram biography
    
    • He began working on probability and numerical analysis, two topics whose practical applications in his day to day work in calculating insurance made their study important to him.
      
    • We have already mentioned the very practical applications to forestry which he continued to study at this time, and his work on probability and numerical analysis involved both the theory and its application to very practical situations.
      
    • Gram's work on probability and numerical analysis led him in a natural way to study abstract problems in number theory.
      

69. Gupta biography
    
    • The authors then investigate how, given a fixed probability p, one can choose a subset of the n populations so that the probability is at least p that the subset contains all those at least as good as the standard population.
      
    • The first of these is Multiple decision procedures: theory and methodology of selecting and ranking populations written jointly with S Panchapakesan and published in the Wiley Series in Probability and Mathematical Statistics in 1979.
      

70. Black Fischer biography
    
    • Inter alia, Bachelier, had shown in his thesis [The Random Character of Stock Market Prices, MIT Press, Cambridge, Massachusetts (contains the translation from French of Bachelier\'s doctoral thesis and contains Sprenkle\'s, 1961 paper).',88)">88] the close connection between random walks and the Fourier heat equation, something that was expanded on by Kac, in 1951, [Ito\'s stochastic calculus and probability theory, Tokyo, ix-xiv.
      
    • Harrison, Krebs, Pliska [Journal for Economic Theory, 20(3), 381-408.',90)">90], [Stochastic Processes and their Applications, 11, 215-260',91)">91] and [Stochastic Processes and their Applications, 15, 313-316.',92)">92] showed that the Black-Scholes formula could be derived using an approach derived from probability theory and the theory of martingales.
      
    • For example, if the traded portfolio was taken as the T-bond, price PT(t), then, if a traded portfolio could replicate the option, then the ratio of the option price O(t) to PT(t) had to be a martingale process under the probability measure induced by PT(t) [Note 13], if there was to be no arbitrage in financial markets.
      

71. Kendall biography
    
    • Kendall is a leading world authority on applied probability and data analysis.
      
    • An exceptional lecturer, Kendall has been the Larmor Lecturer at the Cambridge Philosophical Society in 1980, the Milne Lecturer at Wadham College, Oxford in 1983, the Hoteling Lecturer at the University of North Carolina in 1985, the Rietz Lecturer at the Institute of Mathematical Statistics in 1989 and the Kolmogorov Lecturer at the Bernoulli Society for Mathematical Statistics and Probability in 1990.
      
    • He has also been president of the Bernoulli Society for Mathematical Statistics and Probability in 1975 and of the Mathematics and Physics Sections of the British Association in 1982.
      

72. Cardan biography
    
    • Cardan's understanding of probability meant he had an advantage over his opponents and, in general, he won more than he lost.
      
    • In addition to Cardan's major contributions to algebra he also made important contributions to probability, hydrodynamics, mechanics and geology.
      
    • Cardan makes the first ever foray into the, until then untouched, realm of probability theory.
      

73. Wolf biography
    
    • Wolf wrote on prime number theory and geometry, then later on probability and statistics - a series of papers discussed Buffon's needle experiment in which he estimated π by Monte Carlo methods.
      
    • He published papers in 1849 and 1850 on experiments to compare the experienced probability and the mathematical probability in Versuche zur Vergleichung der Erfahrungswhrscheinlichkeit mit Mathematischen Wahrscheinlichkeit.
      

74. Salem biography
    
    • Another direction in which [Salem] did a lot was applications of the calculus of probability to Fourier series and, curiously enough, this has connection with problems of uniqueness.
      
    • Moreover, it seem that, far from being incidental, as it might have appeared some 30 or so years ago, the calculus of probability is becoming a standard method of attacking problems of trigonometric series.
      
    • Going through the papers of [Salem] one sees this growing role of the calculus of probability.
      

75. Peirce Charles biography
    
    • He wrote on probability arguing against De Morgan's ideas that probability is a measure of confidence and also arguing against the ideas of Bayes.
      
    • Rather for Peirce probability is the limit of the ratio of observed occurrences over the possible occurrences and the number of observations tend to infinity.
      

76. Hopf Eberhard biography
    
    • While at MIT, Hopf did much of his work on ergodic theory which he published in papers such as Complete Transitivity and the Ergodic Principle (1932), Proof of Gibbs Hypothesis on Statistical Equilibrium (1932) and On Causality, Statistics and Probability (1934).
      
    • In this 1934 paper Hopf discussed the method of arbitrary functions as a foundation for probability and many related concepts.
      

77. Petrovsky biography
    
    • Petrovsky's main mathematical work was on the theory of partial differential equations, the topology of algebraic curves and surfaces, and probability.
      
    • Petrovsky also worked on the boundary value problem for the heat equation and this was applied to both probability theory and work of Kolmogorov.
      

78. Wintner biography
    
    • Wintner published on analysis, number theory, differential equations and probability (with several joint papers with Norbert Wiener).
      
    • These were Lectures on asymptotic distributions and infinite convolutions (1938), Analytical foundations of celestial mechanics (1941), Eratosthenian averages (1943), Theory of measure in arithmetical semigroups (1944), The Fourier transforms of probability distributions (1947), and An arithmetical approach to ordinary Fourier series (1945).
      

79. Bellman biography
    
    • We are informed that a particle is in either state 0 or 1, and we are given initially the probability x that it is in state 1.
      
    • Use of the operation A will reduce this probability to ax, where a is some positive constant less than 1, whereas operation L, which consists in observing the particle, will tell us definitely which state it is in.
      

80. Urbanik biography
    
    • Urbanik then began to mix an interest in topology with measure theory and probability and his 1954 papers show this mix: Sur un probleme de J F Pal sur les courbes continues; Limit properties of homogeneous Markov processes with a denumerable set of states; Sur la structure non-topologique du corps des operateurs; and Quelques theoremes sur les measures.
      
    • ',3)">3] divides Urbanik's research into five different major areas: topology, measure theory and analysis; probability theory; stochastic processes; information theory and theoretical physics; and general algebras.
      

81. Hatvani biography
    
    • In this work Hatvani described the theory of probability, in particular basing his material on Jacob Bernoulli's Ars conjectandi.
      
    • Now, I wish to know the probability of the deserter being Hungarian or Croatian.
      

82. Chernoff biography
    
    • The first six chapters deal with the processing of data (graphical methods, means and variances), probability and random variables, the concept of utility (treated axiomatically following von Neumann) and the comparison of various strategies (Bayes, minimax etc.).
      
    • A list of other topics treated follows: D-optimality and the Kiefer-Wolfowitz equivalence theorem; hypothesis-testing in a treatment which is largely, although not whole-heartedly, decision-theoretical; the large-sample evaluation of risk in terms of the Chernoff bounds (a term not used in the text) and the various information numbers; optimisation of sample size in the case of low-cost experimentation; the sequential probability ratio test, no-overshoot approximations, optimality; the Chernoff "procedure A" for sequential design, and its asymptotic optimality; adjacent hypotheses, and the Schwarz boundaries; testing for the sign of a normal mean, with a general consideration of dynamic programming ideas, and of helpful asymptotics; some discussion of one- and two-armed bandits.
      

83. Bernoulli Nicolaus(I) biography
    
    • Five years later he was received a doctorate for a dissertation which studied the application of probability theory to certain legal questions.
      
    • From Montmort's work we can see that Nicolaus formulated certain problems in the theory of probability, in particular the problem which today is known as the St Petersburg problem.
      

84. Hsu biography
    
    • Certainly University College, London was an excellent place for Hsu to study as his mathematical interests were in probability and statistics.
      
    • In 1945 he arrived in the USA just in time for the First Berkeley Symposium on Probability and Statistics.
      

85. Cesaro biography
    
    • the number of common divisors of two numerals, determination of the values of the sum totals of their squares, the probability of incommensurability of three arbitrary numbers, and so on; to these he attempted to apply obtained results in the theory of Fourier series.
      
    • infinite arithmetics, isobaric problems, holomorphic functions, theory of probability, and, particularly, intrinsic geometry.
      

86. Hamming biography
    
    • His major works include Numerical Methods for Scientists and Engineers (1962), Introduction to applied numerical analysis (1971), Digital filters (1977), Coding and information theory (1980), Methods of mathematics applied to calculus, probability, and statistics (1985), Introduction to applied numerical analysis (1989), The Art of Probability for Scientists and Engineers (1991) and The Art of Doing Science and Engineering : Learning to Learn (1997).
      

87. Krylov Nikolai S biography
    
    • he came after the developments in probability theory and ergodic theory of the first three decades of this century which represented the first major impact of statistical mechanics on mathematics (the work of Poincare, Birkhoff, von Neumann, Hopf, Kolmogorov, Khinchin, Wiener, ..
      
    • His works, in which he tried to re-examine the fundamental problems of statistical mechanics in the light of developments in ergodic theory and probability theory, were far ahead of his time.
      

88. Young Lai-Sang biography
    
    • Today it stands at the crossroads of several areas of mathematics, including analysis, geometry, topology, probability, and mathematical physics.
      
    • this implies that the limit theorems of probability hold in this case.
      
    • Her interests include theory, applications and deep connections to mathematical physics and probability.
      

89. Roy biography
    
    • In 1970 the book Essays in probability and statistics, edited by R C Bose, I M Chakravarti, P C Mahalanobis, C R Rao and K J C Smith, dedicated to his memory was published in the University of North Carolina Monograph Series in Probability and Statistics.
      

90. Turing biography
    
    • Turing was elected a fellow of King's College, Cambridge, in 1935 for a dissertation On the Gaussian error function which proved fundamental results on probability theory, namely the central limit theorem.
      
    • Turing's achievements at Cambridge had been on account of his work in probability theory.
      

91. Poincare biography
    
    • In 1886 Poincare was nominated for the chair of mathematical physics and probability at the Sorbonne.
      
    • changing his lectures every year, he would review optics, electricity, the equilibrium of fluid masses, the mathematics of electricity, astronomy, thermodynamics, light, and probability.
      

92. Fisher biography
    
    • The likelihood of a parameter is proportional to the probability of the data and it gives a function which usually has a single maximum value, which he called the maximum likelihood.
      
    • The dispute began in 1917 when Pearson published a paper claiming that Fisher had failed to distinguish likelihood from inverse probability in a paper he wrote in 1915.
      

93. Cournot biography
    
    • Cournot also worked on probability and although his investigations into a logical foundation for it were unsuccessful, his work did lead the way to future important developments.
      
    • He, as Poisson and Condorcet did, applied probability to legal statistics.
      

94. Aitken biography
    
    • The nights were bad, in the daytime colleagues and other friends visited me, and I tried to think about abstract things, such as the theory of probability and the theory of groups - and I did begin to see more deeply into these rather abstruse disciplines.
      
    • From the five minutes of 'stories' one also recalls as part of his lectures on probability a rather stern warning about the evils and foolishness of gambling! .
      

95. Czuber biography
    
    • Among the topics Czuber studied was probability theory and related areas, contributing in 1900 to the Encyklopadie der mathematischen Wissenschaften.
      
    • In fact he wrote the first papers with original results on the probability theory in the Czech region.
      

96. Lame biography
    
    • In the following year he left his chair of physics at the Ecole Polytechnique and accepted a post at the Sorbonne in mathematical physics and probability.
      
    • He was appointed to the chair of mathematical physics and probability at the Sorbonne in 1851.
      

97. Murnaghan biography
    
    • The final book he published was Evaluation of the probability integral to high precision which was a report for the Department of the Navy, David Taylor Model Basin.
      
    • This report describes the determination to high precision of this factor for the asymptotic series representing the probability integral.
      

98. Lacroix biography
    
    • Condorcet was at this time serving as Permanent Secretary to the Academie des Sciences, Inspector General of the Mint, and undertaking a great deal of work on political economy where he applied probability and the philosophy of mathematics.
      
    • Lacroix continued to teach astronomy and the theory of probability at the Lycee.
      

99. Simpson biography
    
    • He also worked on probability theory and in 1740 published The Nature and Laws of Chance.
      
    • In fact he was involved in a dispute with De Moivre over issues of priority on the topic of probability and annuities.
      

100. Ruffini biography
     
     • He also wrote on probability and the application of probability to evidence in court cases.
       

101. Cohen Wim biography
     
     • Using the theory of Markov chains plus some interesting combinatorial manipulations, the author gives explicit formula for the equilibrium probability that a new arrival will meet j busy trunks and the probability that there are j busy trunks at some arbitrary time.
       

102. Geiringer biography
     
     • Her work at this time was on statistics, in particular probability theory, and also on the mathematical theory of plasticity.
       
     • But it may be reasonable to take another standard, that of a university professor of probability and statistics, perhaps an author of the now numerous books on statistical methods.
       

103. Ehrenfest-Afanassjewa biography
     
     • It consists of a critical discussion of the foundations of (classical) statistical mechanics, in particular, the use of the concept 'probability', Boltzmann's H-theorem, the objections of Loschmidt and Zermelo and the various attempts to overcome them, and the difference between the approaches of Boltzmann and Gibbs.
       
     • She also continued to publish works such as Die Grunglagen der Thermodynamik (1956) and On the use of the notion of 'probability' in physics published in the American Journal of Physics in 1957.
       

104. Kendall Maurice biography
     
     • In 1963 he published (jointly with P A P Moran) Geometrical probability followed by Time series (1973) in which Kendall states his objectives to bridge the gap between "sophisticated theory and practical applications" in the field of time series and to "treat the subject in its entirety for the benefit of the practising statistician".
       
     • He also published A course in multivariate analysis and Cluster analysis as well a whole series of articles Studies in the history of probability and statistics.
       

105. Bochner biography
     
     • Again showing wide interests, Bochner worked on the theory of probability.
       
     • His major book on this topic is Harmonic Analysis and the Theory of Probability (1955).
       

106. Price biography
     
     • This is not Laplace's rule of succession, but rather a calculation of the posterior probability that the unknown chance x of the event exceeds 1/2 , based on Bayes's assumption that all values of x are a priori equally likely.
       
     • The Royal Society elected Price a fellow in 1765 on the strength of his contributions to probability.
       

107. Taylor biography
     
     • In particular they discussed infinite series and probability.
       
     • Taylor also corresponded with de Moivre on probability and at times there was a three-way discussion going on between these mathematicians.
       

108. Wiener Norbert biography
     
     • this study introduced me to the theory of probability.
       
     • He introduced a measure in the space of one dimensional paths which brings in probability concepts in a natural way.
       

109. Bell John biography
     
     • If, on the other hand, the initial state-vector has the general form of c+α++ c-α-, then all we can say is that in a measurement of sz, the probability of obtaining the value of planck /2is |c" 2|, and that of obtaining the value of - planck /2is |c-2|.
       
     • (The expectation value of a variable is the mean of the possible experimental results weighted by their probability of occurrence.) Once this mistake was realised, it was clear that hidden variables theories of quantum theory were possible.
       

110. Scott Elizabeth biography
     
     • One particularly interesting result showed, with a probability of error less than 0.02, that when e is greater than 0.12, the distribution of values of the periastron among eclipsing star systems is different from that among the non-eclipsing systems.
       
     • We mention some of the positions she has held: Vice President, American Association for the Advancement of Science, and Chair of the Section on Statistics, 1970-1971; Member of Committee on National Statistics, National Academy of Sciences, 1971-1977; Member of Executive Committee, Caucus for Women in Statistics, 1972-1973 and 1979-1980; Member of the board of Scientific Counsellors, National Institute of Environmental Health Sciences, 1973-1976; President, Institute of Mathematical Statistics, 1977-1978; Member of committee on Education and Employment of women in Science and Engineering, National Academy of Sciences, 1977-1983; Member, Science Indicators Review Task Force, National Science Foundation, 1977-1982; Outstanding Statistician of the Year Award, American Statistical Association, Chicago Chapter, 1980; Honorary Fellow of the Royal Statistical Society, 1981; Vice President, International Statistical Institute, 1981-1983; President, Bernoulli Society for Mathematical Statistics and Probability, International Statistical Institute, 1983-1984.
       

111. Bortkiewicz biography
     
     • Keynes however put the other side of the argument when he wrote in his Treatise on probability (1921):- .
       

112. Einstein biography
     
     • said hardly anything beyond presenting a very simple objection to the probability interpretation ..
       

113. Levy Hyman biography
     
     • Among other mathematical works he published were Numerical Studies in Differential Equations (1934), Elements of the Theory of Probability (1936), and Finite Difference Equations (1958).
       

114. Verhulst biography
     
     • There he gave courses on astronomy, celestial mechanics, the differential and integral calculus, the theory of probability, geometry and trigonometry.
       

115. Wilks biography
     
     • Here he was taught set theory and other courses in advanced mathematics by Robert Moore and he took courses in probability and statistics with E L Dodd.
       

116. Lyapunov biography
     
     • One is certainly his contributions to probability which he became interested in because of courses he was teaching on that subject.
       

117. Hindenburg biography
     
     • Hindenburg published a series of works on combinatorial mathematics, in particular probability, series and formulae for higher differentials.
       

118. Lagrange biography
     
     • His work in Berlin covered many topics: astronomy, the stability of the solar system, mechanics, dynamics, fluid mechanics, probability, and the foundations of the calculus.
       

119. Korteweg biography
     
     • Korteweg showed a similar versatility in his teaching, with his usual courses being analytic and projective geometry, mechanics, astronomy and probability theory.
       

120. Montmort biography
     
     • Montmort's reputation was made by his book on probability Essay d'analyse sur les jeux de hazard which appeared in 1708.
       

121. Anderson biography
     
     • through his origin in the flourishing Russian school of probability, ..
       

122. Planck biography
     
     • from the day I [established a new radiation formula], with the task of finding a real physical interpretation of the formula, and this problem led me automatically to consider the connection between entropy and probability, that is, Boltzmann's train of ideas; eventually after some weeks of the hardest work of my life, light entered the darkness, and a new inconceivable perspective opened up before me.
       

123. Koopmans biography
     
     • His paper The identification of structural characteristics (1950), written jointly with O Reiersol, is concerned with the problem of drawing inferences from the hypothetically exact probability distribution of observed variables to the theoretical structure which generates the distribution.
       

124. Barbier biography
     
     • He also wrote on probability and calculus.
       

125. Pillai biography
     
     • he obtained the probability distributions of statistics relating to several multivariate procedures.
       

126. Pascal biography
     
     • In correspondence with Fermat he laid the foundation for the theory of probability.
       

127. Preston biography
     
     • This was my first experience of research - it was a mixture of algebra and statistics, or probability theory, and I greatly enjoyed it.
       

128. Orlicz biography
     
     • In recent decades those spaces have been used in analysis, constructive theory of functions, differential equations, integral equations, probability, mathematical statistics, etc.
       

129. Minding biography
     
     • At Dorpat Minding taught algebra, analysis, geometry, probability, mechanics and physics.
       

130. Kemeny biography
     
     • A teaching innovation which Kemeny introduced was in developing a Finite Mathematics course including topics that are no surprise to us today: logic, probability and matrix algebra.
       

131. Fine Nathan biography
     
     • To quote just two examples, there is The probability that a matrix be nilpotent written jointly with Herstein and published in the Illinois Journal of Mathematics in 1958, and Pairs of commuting matrices over a finite field written jointly with Walter Feit and published in the Duke Mathematical Journal in 1960.
       

132. Puri biography
     
     • His work in nonparametric statistics and probability theory has had profound effects on the way statistics is understood and applied.
       

133. Bugaev biography
     
     • In this work Bugaev describes mathematics as based on the theory of functions, with geometry and the theory of probability having a minor role.
       

134. Hadamard biography
     
     • He also took up new topics, writing several papers on probability theory, in particular on Markov chains.
       

135. Karlin biography
     
     • to bridge the gap between basic probability know-how and an intermediate level course in stochastic processes, for example, 'A first course in stochastic processes' by the present authors.
       

136. Lexis biography
     
     • Many scientists attempted to adapt probability-based methods to social science problems, including Quetelet and Lexis, but in the end they were frustrated, Quetelet because his methods were too insensitive to segregate his data into categories amenable to statistical analysis, Lexis because his binomial models were insufficiently rich for interesting applications.
       

137. Bernoulli Nicolaus(II) biography
     
     • Nicolaus worked on curves, differential equations and probability.
       

138. Ore biography
     
     • One may add parenthetically that he was occasionally interested himself in this aspect of applied probability.
       

139. Cassini de Thury biography
     
     • Had a proper statistical theory of errors been developed at that time, it would have been possible to give precise levels of probability to the theories and the flattening hypothesis would have been essentially confirmed.
       

140. Martin biography
     
     • In his writings and problem-solving, Martin dealt mostly with Diophantine analysis, probability, elliptic integrals, logarithms, and properties of numbers and triangles.
       

141. Hunt biography
     
     • Hunt then published An inequality in probability theory (1955) and a number of papers in 1956.
       

142. Kramer biography
     
     • Her examination of Omar Khayyam and algebra, Newton and calculus, Fermat and probability, Lewis Carroll and logic and Einstein and relativity provides an intriguing book for non-mathematicians and a valuable reference source for teachers and students.
       

143. Huygens biography
     
     • He informed the mathematicians in Paris including Boulliau of his discovery and in turn Huygens learnt of the work on probability carried out in a correspondence between Pascal and Fermat.
       

144. Dionis biography
     
     • The first was Essai sur les cometes en general; et particulierement sur celles qui peuvent approacher de l'orbite de la terre (1775) which, as the title suggests considers comets and, in particular, shows that the probability of a collision between a comet and the earth is very low.
       

145. Schutzenberger biography
     
     • He found there an incredible interplay between algebra through the use of finite semigroups, probability theory and combinatorics.
       

146. Mackey biography
     
     • This happened when Unitary group representations in physics, probability, and number theory was published in 1978.
       

147. Oresme biography
     
     • Buridan was a philosopher and logician who made contributions to probability, optics and mechanics.
       

148. Cramer biography
     
     • He published an article on the aurora borealis in the Philosophical Transactions of the Royal Society of London and he also wrote an article on law where he applied probability to demonstrate the significance of having independent testimony from two or three witnesses rather than from a single witness.
       

149. Watson Henry biography
     
     • In addition to these books he wrote on Lagrange's method and Monge's method for solving partial differential equations and, jointly with Galton, he wrote On the probability of extinction of families.
       

150. Davidov biography
     
     • As well as his work on the equilibrium of a floating body, Davidov also worked on partial differential equations, elliptic functions and the application of probability to statistics.
       

151. Finsler biography
     
     • At Zurich, in addition to his work on set theory he also worked on differential geometry, number theory, probability theory and the foundations of mathematics.
       

152. Hammersley biography
     
     • Also, anyone who reads the book will, with probability one, become interested in the subject, and will be able to begin to use it.
       

153. Kuczma biography
     
     • Fundamental notions such as existence and uniqueness of solutions of equations under consideration are treated throughout the book as well as a surprisingly wide scale of examples showing applications of the theory in dynamical systems, ergodic theory, functional analysis, functional equations in several variables, functional inequalities, geometry, iteration theory, ordinary differential equations, partial differential equations, probability theory and stochastic processes.
       

154. Wright Sewall biography
     
     • He derives differential equations which are satisfied by the probability density function of the distribution of gene frequencies under certain conditions.
       

155. Walsh Joseph biography
     
     • The topics he taught, rotating them from year to year, included calculus, algebra, mechanics, differential equations, complex variable, probability, number theory, potential theory, approximation theory, and function theory.
       

156. Stokes biography
     
     • I have strongly advised your brother to enter you at Trinity, as I feel convinced that you will in all human probability succeed in obtaining a Fellowship at that College.
       

157. Weldon biography
     
     • Realising that his mathematical skills were somewhat less than he wished, Weldon read widely studying, in particular, the leading works by the French mathematicians on the calculus of probability.
       

158. Wolfowitz biography
     
     • Again he collaborated with Wald on work in this area, and one particular result should be mentioned, namely their proof of the optimal character of the sequential probability ratio test for testing between two hypotheses.
       

159. Vashchenko biography
     
     • In particular he worked on the theory of linear differential equations, the theory of probability (see [A N Bogolyubov (ed.), On the history of the mathematical sciences 167 \'Naukova Dumka\' (Kiev, 1984), 36-39.',3)">3]) and non-euclidean geometry.
       

160. De Witt biography
     
     • He wrote The Worth of Life Annuities Compared to Redemption Bonds which applied probability to questions of state finance.
       

161. Tricomi biography
     
     • These papers cover a vast range of subjects including singular integrals, differential and integral equations, pseudodifferential operators, functional transforms, special functions, probability theory and its applications to number theory.
       

162. Mazur biography
     
     • Mazur was a close collaborator with Banach at Lvov and became a member of the Lvov School of Mathematics, a group of about a dozen mathematicians working in functional analysis, real functions and probability theory.
       

163. Neile biography
     
     • He was a virtuous, sober pious man, and had such a powerful genius to mathematical leaning that had he not been cut off in the prime of his years, in all probability he would have equalled, if not excelled, the celebrated men of that profession.
       

164. Hausdorff biography
     
     • One such lecture course was given on probability theory by Hausdorff in Bonn in the summer of 1923.
       

165. Nash-Williams biography
     
     • In the second of these two papers Nash-Williams considers a recurrent graph, namely one in which if you start at any vertex and move at random to an adjacent vertex then you will return eventually to the starting vertex with probability 1.
       

166. Thiele biography
     
     • [he suggested] consideration of the results of a vote as an outcome of a repeated experiment with fixed probability of distribution, and taking into account the uncertainty due to the sampling when evaluating whether the vote is decisive or not.
       

167. Dantzig biography
     
     • After the Second World War, van Dantzig changed topics and worked on probability and statistics.
       

168. Aristaeus biography
     
     • But one (to us) ordinary property, the focus directrix property, was, as it seems to me, in all probability included.
       

169. Dedekind biography
     
     • Dedekind was then qualified as a university teacher and he began teaching at Gottingen giving courses on probability and geometry.
       

170. Rejewski biography
     
     • Cryptology, that is the science of ciphers, has from the very beginning applied some mathematical methods, mainly the elements of probability theory and statistics.
       

171. Krein biography
     
     • Krein brought the full force of mathematical analysis to bear on problems of function theory, operator theory, probability and mathematical physics.
       

172. Parry biography
     
     • In 1963 he published An ergodic theorem of information theory without invariant measure generalising the individual version of McMillan's ergodic theorem of information theory without the hypothesis of an invariant probability function.
       

173. Wald biography
     
     • The optimum property of the sequential probability ratio test was conjectured by Wald in 1943 and, in a joint paper with Wolfowitz in 1948, he proved this property.
       

174. Malfatti biography
     
     • His papers dealt with many subjects from probability to mechanics and he participated in the debate around Ruffini's attempt to prove the impossibility of solving (in the meaning of that period) equations of higher degree than four.
       

175. Slutsky biography
     
     • Certainly he worked and published on the foundations of probability theory which was a safe political topic.
       

176. Abbe biography
     
     • M G Kendall, see [Biometrika 58 (1971), 369-373.',2)">2] or [Studies in the History of Statistics and Probability II (London, 1977), 331-335.',3)">3], writes:- .
       

177. Pacioli biography
     
     • In [Sciences of the Renaissance (Paris, 1973), 93-106.',10)">10] the importance of Pacioli's work is discussed, in particular his computation of approximate values of a square root (using a special case of Newton's method), his incorrect analysis of certain games of chance (similar to those studied by Pascal which gave rise to the theory of probability), his problems involving number theory (similar problems appeared in Bachet's compilation), and his collection of many magic squares.
       

178. Mittag-Leffler biography
     
     • Mittag-Leffler made numerous contributions to mathematical analysis particularly in areas concerned with limits and including calculus, analytic geometry and probability theory.
       

179. Sommerfeld biography
     
     • He lectured on a wide range of topics, giving lectures on probability and also on the partial differential equations of physics.
       

180. Ampere biography
     
     • This research resulted in him composing a treatise on probability, The Mathematical Theory of Games, which he submitted to the Paris Academy in 1803.
       

181. Jeans biography
     
     • four dimensional space, a space which expands forever; a sequence of events which follows the laws of probability instead of the laws of causation; all these concepts seem to my mind to be structures of pure thought.
       

182. Boltzmann biography
     
     • Boltzmann worked on statistical mechanics using probability to describe how the properties of atoms determine the properties of matter.
       

183. Dandelin biography
     
     • Dandelin also worked on stereographic projection of a sphere on a plane (1827), statics, algebra and probability.
       

184. Besicovitch biography
     
     • He graduated from St Petersburg in 1912 and, influenced by Markov, published his first paper on probability theory.
       

185. Bienayme biography
     
     • After the revolution of 1848 Bienayme retired from the Civil service and he was appointed professor of probability at the Sorbonne.
       

186. Boole biography
     
     • Boole also worked on differential equations, the influential Treatise on Differential Equations appeared in 1859, the calculus of finite differences, Treatise on the Calculus of Finite Differences (1860), and general methods in probability.
       

187. Hudson biography
     
     • Already while at West Ham Institute she had worked on applied probability problems, and now while working for the Air Ministry she published two papers in 1920, one on The strength of lateral loaded struts in The Aeroplane, the other on Incidence wires in the Aeronautical Journal.
       

188. Venn biography
     
     • In 1862 he returned to Cambridge University as a lecturer in Moral Science, studying and teaching logic and probability theory.
       

189. Fermi biography
     
     • Fermi submitted his doctoral thesis Un teorema di calcolo delle probabilita ed alcune sue applicazioni (A theorem on probability and some of its applications) to the Scuola Normale Superiore and was examined on 7 July 1922.
       

190. Lhuilier biography
     
     • He also wrote four important articles on probability during the years 1796 and 1797.
       

191. Poretsky biography
     
     • He applied his logic calculus to the theory of probability.
       

192. Jourdain biography
     
     • This is hardly the place to describe in detail his other activities, such as his editorship of "The Monist" and the International Journal of Ethics, his many researches into the history of science, and his important work on induction and probability, which was in course of publication in "Mind".
       

193. Wrinch biography
     
     • Clearly these lectures made a considerable impression on her forshe published papers in 1919 and 1921 along similar lines of probability as a logical relation of a conclusion to premises.
       

194. Faber biography
     
     • In addition to his research areas, Faber lectured on complex analysis, probability theory, the theory of relativity and analytical mechanics.
       

195. Bronowski biography
     
     • He had presented a series for BBC television in the early 1960s called Insight in which he had looked at mathematical ideas such as probability, scientific ideas such as entropy and also the extent of human intelligence.
       

196. Van Vleck biography
     
     • A brief description of the evolution of the link between measure theory and probability theory is given.
       

197. Clausius biography
     
     • Clausius's great legacy to physics is undoubtedly his idea of the irreversible increase in entropy, and yet we find no indication of interest in Josiah Gibbs' work on chemical equilibrium or Boltzmann's views on thermodynamics and probability, both of which were utterly dependent on his idea.
       

198. Edgeworth biography
     
     • They were applied to the measure of utility, the measure of ethical value, the measure of evidence, the measure of probability, the measure of economic value, and the determination of economic equilibria.
       

199. Montroll biography
     
     • At the Third Berkeley Symposium on Mathematical Statistics and Probability 1954-1955, Montroll gave a paper Theory of the vibration of simple cubic lattices with nearest neighbor interactions in which described vibrations of a cubic lattice with 1, 2, 3, and n dimensions where n is large.
       

200. Wilson Edwin biography
     
     • Not long after the publication of his important text on Aeronautics his interests moved again, this time towards probability and statistics.
       

201. Rogers James biography
     
     • Wiley, New York 1984).',3)">3], [Real Analysis and Probability (Wadsworth, 1989).',5)">5], [Math.
       

202. Maxwell biography
     
     • This theory meant a change from a concept of certainty, heat viewed as flowing from hot to cold, to one of statistics, molecules at high temperature have only a high probability of moving toward those at low temperature.
       

203. Subbotin biography
     
     • Subbotin's early work was in the theory of functions and probability.
       

204. Todhunter biography
     
     • Among his books on the history of mathematics are A History of the Mathematical Theory of Probability from the Time of Pascal to that of Laplace (1865, reprinted 1965) and History of the Mathematical Theories of Attraction (1873).
       

205. Kronecker biography
     
     • students to hear that Kronecker was questioned at his oral on a wide range of topics including the theory of probability as applied to astronomical observations, the theory of definite integrals, series and differential equations, as well as on Greek, and the history of philosophy.
       

206. Birkhoff biography
     
     • This theory, which resolved in principle one of the fundamental problems arising in the theory of gases and statistical mechanics, has been influential not only in dynamics itself but also in probability theory, group theory, and functional analysis.
       

207. Weaver biography
     
     • For example he wrote Lady Luck in 1962 giving a popular account of the theory of probability.
       

208. Hudde biography
     
     • Using the theories of probability and demography being developed by his contemporaries Huygens and Graunt, Hudde gathered data from the records of a previous annuity issued in Amsterdam during the years 1586-90.
       

209. Northcott biography
     
     • Subsequently the department split into four departments: Pure Mathematics, Applied Mathematics, Probability and Statistics, and Computer Science.
       

210. Van der Waerden biography
     
     • Van der Waerden worked on algebraic geometry, abstract algebra, groups, topology, number theory, geometry, combinatorics, analysis, probability theory, mathematical statistics, quantum mechanics, the history of mathematics, the history of modern physics, the history of astronomy and the history of ancient science.
       

211. Christiansen biography
     
     • His first book Elementaer kombinatorik og sandsynlighedsregning (1964) developed the theory of combinatorics and probability and was aimed at school teachers.
       

212. Halmos biography
     
     • This was awarded in 1938 for his thesis on measure-theoretic probability Invariants of Certain Stochastic Transformation: The Mathematical Theory of Gambling Systems.
       

213. Krawtchouk biography
     
     • Other areas he wrote on included algebra (where among other topics he studied the theory of permutation matrices), geometry, mathematical and numerical analysis, probability theory and mathematical statistics.
       

214. Roth biography
     
     • In addition we should mention Roth's other books: Elements of probability (1936), written with Hyman Levy, and Modern elementary geometry (1948).
       

215. Piaggio biography
     
     • Here list a few articles which Piaggio published in The Mathematical Gazette: Relativity rhymes with a mathematical commentary (January 1922); Geometry and relativity (July 1922); Mathematics for evening technical students (July 1924); Mathematical physics in university and school (October 1924); Probability and its applications (July 1931); Three Sadleirian professors: A R Forsyth, E W Hobson and G H Hardy (October 1931); Mathematics and psychology (February 1933); Lagrange's equation (May 1935); Fallacies concerning averages (December 1937); and The incompleteness of "complete" primitives of differential equations (February 1939).
       

216. Rota biography
     
     • The topics were wide-ranging: differential equations, ergodic theory, nonstandard analysis, probability, and of course, combinatorics.
       

217. Braikenridge biography
     
     • Also Concerning the method of constructing a table for the probabilities of life in London, a work involving probability and statistics, as did two other works Concerning the number of people in England and Concerning the present increase of the people in Britain and Ireland.
       

218. Heisenberg biography
     
     • He did not invent these concepts as a matrix algebra, however, rather he focused attention on a set of quantised probability amplitudes.
       

219. Bernoulli Johann(III) biography
     
     • In the field of mathematics he worked on probability, recurring decimals and the theory of equations.
       

History Topics

1. references
   
   • (1942), On stochastic processes (Infinitely divisible laws of probability), Japanese Journal of Mathematics.
     
   • (1984), Introduction to Probability Theory, Cambridge University Press (translated from the Japanese), .
     
   • (1996), in Ikeda N, Watanabe S, Fukushima M and Kunita H (eds.), It™'s stochastic calculus and probability theory, Tokyo, ix-xiv.
     
   • (1951), On some connections between probability theory and differential and integral equations, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, 189-215, University of California Press.
     

2. Voting
   
   • Llull's idea of a fair election was proposed again 500 years later by Condorcet in his Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix (Essay on the Application of Analysis to the Probability of Majority Decisions) published in 1785.
     
   • This is a remarkable work which gives Condorcet an important place in the history of probability.
     

3. Real numbers 3
   
   • Pick a real at random, and the probability is zero that it's accessible - the probability is zero that it will ever be accessible to us as an individual mathematical object.
     

4. Wave versus matrix
   
   • Heisenberg invented these concepts by focusing attention on a set of quantised probability amplitudes.
     

5. History overview
   
   • Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability.
     Go directly to this paragraph
     

6. Pi history
   
   • If we have a uniform grid of parallel lines, unit distance apart and if we drop a needle of length k < 1 on the grid, the probability that the needle falls across a line is 2k/π.
     Go directly to this paragraph
     

7. Statistics index
   
   • Bertrand Introduction to Bertrand's work on probability .
     

8. Neptune and Pluto
   
   • .the extreme probability of now discovering a new planet in a very short time, provided the powers of one observatory could be directed to search for it.
     

9. Pi history references
   
   • N T Gridgeman, Geometric probability and the number π, Scripta Math.
     

10. Orbits references
    
    • B Gower, Planets and probability : Daniel Bernoulli on the inclinations of the planetary orbits, Stud.
      

11. Pi history references
    
    • N T Gridgeman, Geometric probability and the number π, Scripta Math.
      

12. Modern light
    
    • Rayleigh (and Jeans) also tried to explain Wien's blackbody result making the assumption that all possible frequency modes could radiate with equal probability.
      

13. Tait's scrapbook
    
    • If the Earl of March were created a May-Duke determine the probability of his becoming a Winter-King and if he were elected Emperor would it be right to address him as Semper Augustus.
      

14. Orbits references
    
    • B Gower, Planets and probability : Daniel Bernoulli on the inclinations of the planetary orbits, Stud.
      

Famous Curves

Societies etc

1. AMS Steele Prize
   
   • for his paper "Ergodic theory and its significance for statistical mechanics and probability theory".
     
   • for his cumulative influence on the fields of probability theory, Fourier analysis, several complex variables, and differential geometry.
     
   • for his fundamental work in establishing probability as a branch of mathematics and for his continuing profound influence on its development.
     
   • for his foundational contributions to Lie algebras and probability theory over a long period and his production of outstanding research students in both Russia and the United States, countries to whose mathematical life he has contributed so richly.
     
   • In these papers he showed for the first time how to use the powerful tools of probability theory to attack the hard analytic questions of constructive quantum field theory, controlling renormalizations with Lp estimates in the first paper, and in the second turning Euclidean quantum field theory into a subset of the theory of stochastic processes.
     
   • for his many and deep contributions to probability theory and its applications.
     
   • for his rich and magnificent mathematical career and for his work in analysis, which has a strong orientation towards probability theory.
     

2. International Congress Speakers
   
   • William Feller, Some New Connections between Probability and Classical Analysis.
     
   • Albert Nikolayevich Shiryaev, Absolute Continuity and Singularity of Probability Measures in Functional Spaces.
     
   • Aleksander Pelczynski, Structural Theory of Branch Spaces and Its Interplay with Analysis and Probability.
     
   • Dan Voiculescu, Free Probability Theory: Random Matrices and von Neumann Algebras.
     
   • Uffe Haagerup, Random Matrices, Free Probability and the Invariant Subspace Problem Relative to a von Neumann Algebra.
     

3. BMC 1986
   
   • Barnett, C N on-commutative probability - limit theorems and maximal inequalities .
     
   • Streater, R F Classical and quantum probability .
     

4. Wilks Award of the ASS
   
   • for outstanding research in Time Series Analysis, especially for his innovative introduction of reproducing kernel spaces, spectral analysis and spectrum smoothing; for pioneering contributions in quantile and density quantile functions and estimation; for unusually successful and influential textbooks in Probability and Stochastic Processes; for excellent and enthusiastic teaching and dissemination of statistical knowledge; and for a commitment to service on Society Councils, Government Advisory Committees, and Editorial Boards.
     
   • for path breaking research contributions in the applications of sophisticated probability tools to statistical methodology and scientific applications, particularly on optimal stopping, sequential analysis, change-point problems and genetic linkage.
     
   • for extraordinary broad and deep contributions to applied statistics methodology, to mathematical statistics, and to probability, encompassing topics such as regression model selection, covering designs, rankings, graphics, combinatorics, coding theory, and the foundations of data analysis; and for generous, unstinting and productive collaborations and guidance to other statisticians, mathematicians, scientists, engineers, and business executives.
     

5. Fermat Prize
   
   • The Fermat Prize is awarded to a mathematician for decisive research in those fields to which Pierre de Fermat contributed, namely: Statements of Variational Principles; Foundations of Probability and Analytical Geometry; and Number Theory.
     
   • for his fundamental contributions in various domains of Probability.
     

6. European Mathematical Society Prizes
   
   • Recently he proved a remarkable "Optional decomposition of supermartingales" which is an extension of the fundamental Doob-Meyer decomposition for the case of many probability measures.
     
   • has became known through his results on Probability theory.
     

7. Bulgarian Statistical Society
   
   • Before this there had been a series of Summer Schools on Probability and Statistics organised jointly by the Bulgarian Academy of Sciences and Sofia University.
     
   • We mentioned above the Summer School on Probability and Statistics.
     

8. Bulgarian Academy of Sciences
   
   • A second example is the Sixth International Summer School on Probability Theory and Mathematical Statistics took place at the house of the Bulgarian Academy of Sciences in Varna (Golden Sands) from 28 September to 10 October 1988.
     
   • The Summer School was organised by the Department of Probability and Statistics of the Institute of Mathematics.
     

9. References for Russian
   
   • A N Shiryaev, On the history of the founding of the Russian Academy of Sciences and on the first publications on probability theory in Russian editions (Russian), Teor.
     
   • A N Shiryaev, On the history of the founding of the Russian Academy of Sciences and on the first publications on probability theory in Russian editions, Theory Probab.
     

10. Minutes for 1954
    
    • It was agreed that one day should be devoted to each of Probability with Logic and Foundations, Analysis with Number Theory, and Topology.
      

11. BMC 1989
    
    • Cutland, N J Applications of non-standard analysis to probability theory .
      

12. BMC 1999
    
    • Shalev, A Simple groups, Cayley groups and probability .
      

13. MAA Chauvenet Prize
    
    • The Foundations of Probability, Amer.
      

14. Vietnamese Mathematical Society
    
    • It was the Vietnamese Mathematical Society which continued to support mathematical research with regularly organised seminars in optimisation, probability, functional analysis, algebra, and numerical analysis.
      

15. SIAM George Pólya Prize
    
    • for a notable contribution in another area of interest to George Polya such as approximation theory, complex analysis, number theory, orthogonal polynomials, probability theory, or mathematical discovery and learning.
      

16. NAS Award in Mathematics
    
    • for the theory of free probability, in particular, using random matrices and a new concept of entropy to solve several hitherto intractable problems in von Neumann algebras.
      

17. NAS Award in Applied Mathematics
    
    • for his brilliant and productive mathematical work encompassing genetics, economics, approximation theory, probability, and statistics, and game theory.
      

18. Minutes for 1977
    
    • 5 Analysis, 3 Topology, 4 Algebra, 2 Number Theory, 1 Geometry, 1 Combinatorics, 1 Logic, 1 Probability.
      

19. AMS/SIAM Birkhoff Prize
    
    • for his important contributions to statistical mechanics and to probability theory and its applications.
      

20. Abel Prize
    
    • for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.
      

21. BMC 1978
    

22. BMC 1982
    
    • Stone, MHazards of infinity in probability and statistics .
      

23. Czech Academy of Sciences
    
    • The Institute is concerned mainly with mathematical analysis (differential equations, numerical analysis, functional analysis, theory of functions, mathematical physics), probability theory and mathematical statistics, mathematical logic, theoretical computer science and graph theory, numerical algebra, topology (general and algebraic) and theory of teaching mathematics.
      

24. BMC 1969
    
    • Offord, A CA survey of some applications of the theory of probability in analysis .
      

25. Minutes for 1989
    
    • J Hawkes (0) Swansea Probability .
      

26. BMC 1956
    
    • Offord, A CThe theory of probability in analysis .
      

27. BMC 1985
    
    • Hammersley, J M Probability and arithmetic in science .
      

References

1. References for Laplace
   
   • F J Giron, History of probability theory: from Pascal to Laplace , in History of mathematics in the XIXth century (Spanish) 2 (Madrid, 1994), 113-133.
     
   • M A Gomez Villegas, The problem of inverse probability : Bayes and Laplace (Spanish), in Current perspectives in logic and philosophy of science (Spanish) (Madrid, 1994), 385-396.
     
   • A Kamlah, The decline of the Laplacian theory of probability : a study of Stumpf, von Kries, and Meinong, in The probabilistic revolution 1 (Cambridge, MA-London, 1987), 91-116.
     
   • I Schneider, Laplace and thereafter : the status of probability calculus in the nineteenth century, in The probabilistic revolution 1 (Cambridge, MA-London, 1987), 191-214.
     
   • O B Sheynin, The probability theory of P-S Laplace (Russian), Istor.-Mat.
     
   • O B Sheynin, P S Laplace's work on probability, Arch.
     
   • S M Stigler, Laplace's 1774 memoir on inverse probability, Statist.
     
   • S M Stigler, Studies in the history of probability and statistics XXXIV: Napoleonic statistics : the work of Laplace, Biometrika 62 (2) (1975), 503-517.
     
   • S M Stigler, Studies in the history of probability and statistics XXXII : Laplace, Fisher, and the discovery of the concept of sufficiency, Biometrika 60 (1973), 439-445.
     

2. References for Renyi
   
   • N H Bingham, The work of Alfred Renyi: some aspects in probability and number theory, Studia Sci.
     
   • D G Kendall, Obituary: Alfred Renyi, Journal of Applied Probability 7 (1970), 509-522.
     
   • The work of Alfred Renyi in probability theory (Hungarian), Mat.
     
   • L Schmetterer, Alfred Renyi, in memoriam, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability Vol.
     
   • P Turan, The life of Alfred Renyi (1921-1970), Department of Probability Theory and Statistics, Eotvos Lorand University.
     
   • http://www.math.elte.hu/probability/renyi/racv.html .
     

3. References for Black Fischer
   
   • (1942), On stochastic processes (Infinitely divisible laws of probability), Japanese Journal of Mathematics.
     
   • (1984), Introduction to Probability Theory, Cambridge University Press (translated from the Japanese), .
     
   • (1996), in Ikeda N, Watanabe S, Fukushima M and Kunita H (eds.), Ito's stochastic calculus and probability theory, Tokyo, ix-xiv.
     
   • (1951), On some connections between probability theory and differential and integral equations, Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, 189-215, University of California Press.
     

4. References for Levy Paul
   
   • L Le Cam, Paul Levy, 1886-1971, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability III: Probability theory (Berkeley, Calif., 1972), xiv-xx.
     
   • J L Doob, Paul Levy, Annals of Probability 1 (1971), 1-18.
     
   • Probability 9 (1972), 870-872.
     
   • Probability 1 (1) (1971), 1-18.
     

5. References for Bernoulli Daniel
   
   • B Gower, Planets and probability : Daniel Bernoulli on the inclinations of the planetary orbits, Stud.
     
   • O B Sheynin, Daniel Bernoulli on the normal law, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 101-104.
     
   • O B Sheynin, D Bernoulli's work on probability, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 105-133.
     
   • O B Sheynin, D Bernoulli's work on probability, RETE 1 (3-4) (1971/72), 273-300.
     

6. References for Cantelli
   
   • M Benzi, Uncertainties and controversies: the debate about logic and probability in Italy in the early twentieth century (Italian), Historia Math.
     
   • G Fenaroli, F Furinghetti and A M Somaglia, Works on probability theory published in Italian journals in the late nineteenth and early twentieth centuries (Italian), in Conference on the History of Mathematics, Cetraro, 1988 (EditEl, Rende, 1991), 105-141.
     
   • E Regazzini, Probability theory in Italy between the two world wars.
     
   • E Regazzini, Probability and statistics in Italy during the First World War I: Cantelli and the laws of large numbers, Electronic Journal for History of Probability and Statistics 1 (1) (2005).
     

7. References for Bertrand
   
   • A I Dale, A history of inverse probability : From Thomas Bayes to Karl Pearson (Second edition), Sources and Studies in the History of Mathematics and Physical Sciences (Springer-Verlag, New York, 1999).
     
   • O B Sheynin, Bertrand's work on probability, Arch.
     
   • O B Sheynin, Geometric probability and the Bertrand paradox, Historia Sci.
     
   • O B Sheynin, H Poincare's work on probability, Arch.
     

8. References for Quetelet
   
   • L J Daston, Rational individuals versus laws of society : from probability to statistics, in The probabilistic revolution 1 (MIT Press, Cambridge, MA-London, 1987), 295-304.
     
   • P F Lazarsfeld, Notes on the history of quantification in sociology - trends, sources and problems, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 213-270.
     
   • B-P Lecuyer, Probability in vital and social statistics : Quetelet, Farr, and the Bertillons, in The probabilistic revolution 1 (MIT Press, Cambridge, MA-London, 1987), 317-335.
     
   • A Oberschall, The two empirical roots of social theory and the probability revolution, The probabilistic revolution 2 (MIT Press, Cambridge, MA, 1987), 103-131.
     

9. References for Boscovich
   
   • C Eisenhart, Boscovich and the combination of observations, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 88-100.
     
   • R W Farebrother, Studies in the history of probability and statistics XLII.
     
   • O B Sheynin, R J Boscovich's work on probability, Arch.
     
   • S M Stigler, Studies in the history of probability and statistics.
     

10. References for Mises
    
    • H Geiringer (ed.), Mathematical Theory of Probability and Statistics (New York, 1964).
      
    • H Cramer, Richard von Mises' work in probability and statistics, Ann.
      
    • M van Lambalgen, Randomness and foundations of probability : von Mises' axiomatisation of random sequences, in Statistics, probability and game theory (Hayward, CA, 1996), 347-367.
      

11. References for Bachelier
    
    • W Feller, An Introduction to Probability Theory and its Applications, Vols.
      
    • M Kac, Probability and Related Topics in Physical Sciences (1959) Interscience, New York.
      
    • K Ito, Introduction to Probability Theory (1984) Cambridge University Press (translated from the Japanese), .
      
    • K Ito in Ikeda N, Watanabe S, Fukushima M and Kunita H (eds.), Ito's stochastic calculus and probability theory (1996) Tokyo, ix-xiv.
      

12. References for Kac
    
    • K Baclawski and M D Donsker (eds.), Mark Kac : probability, number theory, and statistical physics, Mathematicians of Our Time 14 (Cambridge, Mass.-London, 1979).
      
    • H Kesten, The influence of Mark Kac on probability theory, Ann.
      
    • H Kesten, Correction : 'The influence of Mark Kac on probability theory', Ann.
      

13. References for Markov
    
    • A Besicovitch, Markov's works in probability (Russian), Izvestiya Rossiiskoi akademii nauk 17 (1923).
      
    • O B Sheynin, A A Markov's Work on Probability, Archive for History of Exact Science 39 (1988), 337-377.
      
    • O B Sheynin, Errata for the contribution: 'A A Markov's work on probability', Archive for History of Exact Science 40 (4) (1989), 387.
      

14. References for Crofton
    
    • A Dale, A history of inverse probability : From Thomas Bayes to Karl Pearson (New York, 1999).
      
    • M T Peres Larigno, On the history of the notion of geometric probability (Russian), Voprosy Istor.
      
    • E Seneta, K Parshall and F Jongmans, Nineteenth-century developments in geometric probability : J J Sylvester, M W Crofton, J-E Barbier, and J Bertrand, Arch.
      

15. References for Doeblin
    
    • H Cohn (ed.), Doeblin and Modern Probability (Providence, RI, 1993).
      
    • W Feller, An Introduction to Probability Theory and Its Applications Vol.
      
    • B Bru, Doeblin's life and work from his correspondence, in H Cohn (ed.), Doeblin and Modern Probability (Providence, RI, 1993).
      

16. References for Buffon
    
    • P Holgate, Buffon's cycloid, Studies in the history of probability and statistics XXXIX, Biometrika 68 (3) (1981), 712-716.
      
    • P Holgate, Correction : 'Studies in the history of probability and statistics.
      
    • J Roger, Buffon and mathematics, Geometrical probability and biological structure : Buffon's 200th anniversary, in Proc.
      

17. References for Pearson
    
    • J B S Haldane, Karl Pearson, 1857 - 1957 : A centenary lecture delivered at University College London, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 427-438.
      
    • E S Pearson, Some early correspondence between W S Gosset, R A Fisher and Karl Pearson, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 405-418.
      
    • Historical sidelights on an episode in Anglo-Polish collaboration, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 455-479.
      

18. References for Ito
    
    • N Ikeda, S Watanabe, M Fukushima and H Kunita (eds.), Ito's stochastic calculus and probability theory (Tokyo, 1996).
      
    • K Ito, My Sixty Years in Studies of Probability Theory : acceptance speech of the Kyoto Prize in Basic Sciences (1998).
      
    • Kiyosi Ito, in N Ikeda, S Watanabe, M Fukushima and H Kunita (eds.), Ito's stochastic calculus and probability theory (Tokyo, 1996), ix-xiv.
      

19. References for Feller
    
    • J L Doob, William Feller and twentieth century probability, in Proc.
      
    • of Sixth Berkeley Conference of Mathematical Statistics and Probability II (Berkeley, 1972), xv-xx.
      
    • of Sixth Berkeley Conference of Mathematical Statistics and Probability II (Berkeley, 1972), xxi-xxiii.
      

20. References for Pascal
    
    • L E Maistrov, Pascal's concept of probability according to A Renyi (Russian), Istor.-Mat.
      
    • O Ore, Pascal and the invention of probability theory, Amer.
      
    • B L Van der Waerden, The correspondence between Pascal and Fermat on questions of probability theory (Russian), Istor.-Mat.
      

21. References for Puri
    
    • E Brunner and M Denker (eds.), Research developments in probability and statistics : Festschrift in honor of Madan L Puri on the occasion of his 65th birthday (VSP, Utrecht, 1996).
      
    • E Brunner and M Denker, Preface dedicated to Madan L Puri, on the occasion of his 65th birthday, in E Brunner and M Denker (eds.), Research developments in probability and statistics : Festschrift in honor of Madan L Puri on the occasion of his 65th birthday (VSP, Utrecht, 1996), xi-xii.
      
    • List of publications of Madan L Puri, in E Brunner and M Denker (eds.), Research developments in probability and statistics : Festschrift in honor of Madan L Puri on the occasion of his 65th birthday (VSP, Utrecht, 1996), xiii-xxviii.
      

22. References for Lukacs
    
    • J Gani and V K Rohatgi (eds.), Contributions to Probability (New York, 1981).
      
    • Applied Probability 25 (1988), 641-646.
      

23. References for Kiefer
    
    • L D Brown, I Olkin, J Sacks and H P Wynn (eds.), Jack Carl Kiefer, Collected papers I : Statistical inference and probability (1951-1963) (New York, 1985).
      
    • L D Brown, I Olkin, J Sacks and H P Wynn (eds.), Jack Carl Kiefer, Collected papers II : Statistical inference and probability (1964-1984) (New York, 1985).
      

24. References for Hsu
    
    • K L Chung, Hsu's work in probability, Ann.
      
    • K L Chung, The work of Pao Lu Hsu on probability theory (Chinese), Knowledge Practice Math.
      

25. References for Lambert
    
    • U Garibaldi and M A Penco, Probability theory and physics between Bernoulli and Laplace : the contribution of J H Lambert (1728-1777) (Italian), in Proceedings of the fifth national congress on the history of physics, Rome, 1984, Rend.
      
    • O B Sheynim, J H Lambert's Work on Probability, Archive for History of Exact Sciences 7 (1971), 244-256.
      

26. References for Edgeworth
    
    • P J FitzPatrick, Leading British statisticians of the nineteenth century, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 180-212.
      
    • M G Kendall, Francis Ysidro Edgeworth, 1845-1926, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 257-263.
      

27. References for De Finetti
    
    • D M Cifarelli and E Regazzini, De Finetti's contribution to probability and statistics, Statistical Science 11 (1996), 253-282.
      
    • B de Finetti, Probability and my life, in J Gani (ed.), The Making of Statisticians (New York, 1982), 3-12.
      

28. References for Gosset
    
    • E S Pearson, Some early correspondence between W S Gosset, R A Fisher and Karl Pearson, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 405-418.
      
    • William Sealy Gosset, 1876-1937, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 355-404.
      

29. References for Hunt
    
    • K MacPherson, Obituary: Gilbert Hunt - Princeton probability expert, University World News (29 June 2008).
      
    • K MacPherson, Gilbert Hunt, probability expert, dies at 92, Princeton Weekly Bulletin 97 (29) (16 June 2008).
      

30. References for Keynes
    
    • R B Braithwaite (ed.), J M Keynes, The collected writings of John Maynard Keynes VIII : A treatise on probability (New York, 1988).
      
    • J Runde, Keynes after Ramsey: in defence of 'A treatise on probability', Stud.
      

31. References for D'Alembert
    
    • L Daston, d'Alembert's critique of probability theory, Historia Math.
      
    • L E Maistrov, D'Alembert and the theory of probability (Russian), Istor.-Mat.
      

32. References for Neyman
    
    • R Bartoszynski, Some thoughts about Jerzy Neyman, Mathematical statistics and probability theory (New York- Berlin, 1980), ix-xvi.
      
    • K Krickeberg, The role of Jerzy Neyman in the shaping of the Bernoulli Society, Mathematical statistics and probability theory (New York- Berlin, 1980), xx-xxii.
      

33. References for Khinchin
    
    • H Cramer, A I Khinchin's work in mathematical probability, Ann.
      
    • B V Gnedenko, The Department of Probability Theory of Moscow State University (Russian), Teor.
      

34. References for Simpson
    
    • R W Farebrother, Studies in the history of probability and statistics XLII.
      
    • S M Stigler, Studies in the history of probability and statistics.
      

35. References for Bayes
    
    • A Hald, Evaluations of the beta probability integral by Bayes and Price, Arch.
      
    • Thomas Bayes's essay towards solving a problem in the doctrine of chances, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 131-153.
      

36. References for Doob
    
    • K L Chung, Probability and Doob, Amer.
      
    • J L Doob, The development of rigor in mathematical probability (1900-1950), Amer.
      

37. References for Kendall
    
    • D G Kendall, Applied probability - past, present and future : A celebration of applied probability, J.
      

38. References for Chebyshev
    
    • N S Ermolaeva, An unpublished course on probability theory by P L Chebyshev (Russian), Voprosy Istor.
      
    • O Sheynin, Chebyshev's lectures on the theory of probability, Arch.
      

39. References for Ostrogradski
    
    • B V Gnedenko, On M V Ostrogradskii 's works on the theory of probability (Russian), Istor.-Mat.
      

40. References for Babbage
    
    • P J FitzPatrick, Leading British statisticians of the nineteenth century, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 180-212.
      

41. References for Arbuthnot
    
    • M T Peres Larin'o, On the history of the notion of geometric probability (Russian), Voprosy Istor.
      

42. References for Ptolemy
    
    • A T Fomenko, V V Kalashnikov and G V Nosovskii, Statistical analysis and dating of the observations on which Ptolemy's 'Almagest' star catalogue is based, in Probability theory and mathematical statistics I (Vilnius, 1990), 360-374.
      

43. References for Naimark
    
    • D P Zhelobenko, The scientific work of M A Naimark (Russian), Probability theory, function theory, mechanics, Trudy Mat.
      

44. References for Fisher
    
    • E S Pearson, Some early correspondence between W S Gosset, R A Fisher and Karl Pearson, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 405-418.
      

45. References for De Moivre
    
    • R H Daw and E S Pearson, Studies in the history of probability and statistics XXX : Abraham de Moivre's 1733 derivation of the normal curve : a bibliographical note, Biometrika 59 (1972), 677-680.
      

46. References for Abbott
    
    • J Smith, L I Berkove and G A Baker, A grammar of dissent : Flatland, Newman, and the theology of probability, Victorian Studies 39 (1996), 129-150.
      

47. References for Huygens
    
    • H Freudenthal, Huygens' foundations of probability, Historia Math.
      

48. References for Watson Henry
    
    • D G Kendall, Branching processes since 1873, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 383-405.
      

49. References for Craig James
    
    • S M Stigler, John Craig and the probability of history : from the death of Christ to the birth of Laplace, Journal of the American Statistical Association 81 (1986), 879-887.
      

50. References for Steklov
    
    • H O Ondar, A certain work of V A Steklov on probability theory (Russian) History and Methodology of Natural Sciences.
      

51. References for Weldon
    
    • K Pearson, Walter Frank Raphael Weldon, 1860-1906, in E S Pearson and M G Kendall, Studies in the History of Statistics and Probability (London, 1970), 265-322.
      

52. References for Vashchenko
    
    • R A Sapsai, The work of Mikhail Egorovich Vashchenko-Zakharchenko in the field of probability theory (Russian), in A N Bogolyubov (ed.), On the history of the mathematical sciences 167 'Naukova Dumka' (Kiev, 1984), 36-39.
      

53. References for Waring
    
    • P Holgate, Studies in the history of probability and statistics XLI : Waring and Sylvester on random algebraic equations, Biometrika 73 (1) (1986), 228-231.
      

54. References for Hausdorff
    
    • L Olsen, Review of Integral, probability, and fractal measures, by G Edgar (New York, 1998), Bull.
      

55. References for Blackwell
    
    • T S Ferguson, L S Shapley and J B MacQueen (eds.), Statistics, probability and game theory, Papers in honor of David Blackwell, Institute of Mathematical Statistics Lecture Notes - Monograph Series 30 (Hayward, CA, 1996).
      
    • T S Ferguson, L S Shapley and J B MacQueen, Preface [biography of David Blackwell], in Statistics, probability and game theory (Inst.
      

56. References for Newcomb
    
    • S M Stigler, Simon Newcomb, Percy Daniell, and the history of robust estimation 1885-1920, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 410-418.
      

57. References for Price
    
    • A Hald, Evaluations of the beta probability integral by Bayes and Price, Arch.
      

58. References for Sprague
    
    • Sprague wrote actuarial papers on mortality, graduation, interpolation, life interests and reversionary interests, combined marriage and mortality tables, vaccination against smallpox, premium rates, distribution of life office surplus, valuation of life office liabilities, life office expenses, legislation of life companies, causes of life office insolvencies, transfer of business from one life office to another, use of probability in actuarial studies.
      

59. References for Jevons
    
    • P J FitzPatrick, Leading British statisticians of the nineteenth century, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 180-212.
      

60. References for Galton
    
    • P J FitzPatrick, Leading British statisticians of the nineteenth century, in M G Kendall and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 180-212.
      

61. References for Wald
    
    • Selected papers in statistics and probability by Abraham Wald (New York-Toronto-London, 1955).
      

62. References for Abbe
    
    • M G Kendall, The work of Ernst Abbe, in M G Kendal and R L Plackett (eds.), Studies in the History of Statistics and Probability II (London, 1977), 331-335.
      

63. References for Gnedenko
    
    • A N Kolmogorov, On the works of B V Gnedenko in probability theory (Russian), Teor.
      

64. References for Cramer Harald
    
    • E J Wegman, Some personal recollections of Harald Cramer on the development of statistics and probability, Statist.
      

65. References for Euclid
    
    • K Kreith, Euclid turns to probability, Internat.
      

66. References for Poisson
    
    • O B Sheynin, S D Poisson's work in probability, Arch.
      

67. References for Fermat
    
    • B L Van der Varden, The correspondence between Pascal and Fermat on questions of probability theory (Russian), Istor.-Mat.
      

68. References for Linnik
    
    • Probability 10 (1973), 244-246.
      

69. References for Bonferroni
    
    • E Seneta, Probability inequalities and Dunnett's test, In F M Hoppe, Multiple Comparisons, Selections and Applications in Biometry, (New York, 1993) 29-45.
      

70. References for Kolmogorov
    
    • A N Shiryaev et al., Andrei Nikolaevich Kolmogorov : (April 25, 1903 - October 20, 1987) : in memoriam, Theory of probability and its applications 34 (1989), 5-118.
      

71. References for Polya
    
    • K L Chung, Polya's work in probability, Bull.
      

72. References for Rogers James
    
    • U Dudley, Real Analysis and Probability (Wadsworth, 1989).
      

73. References for Steinhaus
    
    • K Urbanik, The ideas of Hugo Steinhaus on probability theory (Polish), Wiadomosci matematyczne (2) 17 (1973), 39-50.
      

74. References for Gateaux
    
    • Daniel W Stroock, Probability theory.
      

75. References for Craig
    
    • S M Stigler, John Craig and the probability of history : from the death of Christ to the birth of Laplace, Journal of the American Statistical Association 81 (1986), 879-887.
      

76. References for Bernoulli Jacob
    
    • E Sylla, Political, moral, and economic decisions and the origins of the mathematical theory of probability : the case of Jacob Bernoulli's 'The art of conjecturing', in Acting under uncertainty : multidisciplinary conceptions (Boston, MA, 1990), 19-44.
      

77. References for Dirichlet
    
    • H Fischer, Dirichlet's Contribution to Mathematical Probability Theory, Historia Mathematica 21 (1994), 39-63.
      

78. References for Savage
    
    • D V Lindley, L J Savage : His work in probability and statistics, Annals of Statistics 8 (1980), 1-24.
      

79. References for Bunyakovsky
    
    • O B Sheynin, On V Ya Buniakovsky's work in the theory of probability, Arch.
      

Additional material

1. Harold Jeffreys on Probability
   
   • Harold Jeffreys on Probability .
     
   • Between 1919 and 1923 Harold Jeffreys and Dorothy Wrinch wrote three papers on probability and scientific inference.
     
   • D Wrinch and H Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 38 (1919), 715-731.
     
   • In their views of probability they were influenced by William Ernest Johnson and John Maynard Keynes.
     
   • We give below an extract from Jeffreys' book on Probability.
     
   • Probability .
     
   • What is probability? .
     
   • Probability expresses a relation between a proposition and a set of data.
     
   • When the data imply that the proposition is true, the probability is said to amount to certainty; when they imply that it is false, the probability becomes impossibility.
     
   • All intermediate degrees of probability can arise.
     
   • The relation of the laws of science to the data of observation is one of probability.
     
   • The more facts are in agreement with the inferences from a law, the higher the probability of the law becomes; but a single fact not in agreement may reduce a law, previously practically certain, to the status of an impossible one.
     
   • Its probability increased as it was shown to fit the motions of the planets, satellites, and comets, and those of double stars, with an astonishing degree of accuracy.
     
   • The fundamental notion of probability is intelligible a priori to everybody, and is regularly used in everyday life.
     
   • This might suggest that probability is a matter of differences between individuals.
     
   • We say the same about probability.
     
   • On a given set of data p we say that a proposition q has in relation to these data one and only one probability.
     
   • If any person assigns a different probability, he is simply wrong, and for the same reasons as we assign in the case of logical judgments.
     
   • Personal differences in assigning probabilities in everyday life are not due to any ambiguity in the notion of probability itself, but to mental differences between individuals, to differences in the data available to them, and to differences in the amount of care taken to evaluate the probability.
     
   • Principles of probability .
     
   • The mathematical discussion of probability depends on the principle that probabilities can be expressed by means of numbers.
     
   • If we have two sets of data p and p', and two propositions q and q', and we consider the probabilities of q given p, and of q' given p', then whatever p, p', q, q' may be, the probability of q given p is either greater than, equal to, or less than that of q' given p'.
     
   • All propositions impossible on the data have the same probability, which is not greater than any other probability; and all propositions certain on the data have the same probability, which is not less than any other probability.
     
   • The relations greater than and less than are transitive; that is, if one probability is greater than a second, and the second greater than a third, then the first probability is greater than the third.
     
   • If one probability is greater than a second, the second is said to be less than the first; and if neither of two probabilities is greater than the other we say that they are equal.
     
   • This postulate ensures the existence of a definite order among probabilities, such that each probability follows all smaller ones and precedes all greater ones.
     
   • Such an order once established, we can construct a correspondence between probabilities and real numbers, so that to every probability corresponds one and only one number, and so that of every pair of probabilities the less corresponds to the smaller number.
     
   • We need a further rule before we can decide what number to attach to any given probability.
     
   • If several propositions are mutually contradictory on the data, the number attached to the probability that some one of them is true shall be the sum of those attached to the probabilities that each separately is true.
     
   • Any rational proper fraction, including 0 and 1, can be a probability number.
     
   • It follows from this that any probability can be made to correspond to a real number, rational or irrational.
     
   • For any given probability P either corresponds to a rational fraction or does not.
     
   • In the latter case every R-probability is either greater or less than P.
     
   • Also, since the relation "greater than" among probabilities is transitive, every fraction corresponding to an R probability is greater than every fraction corresponding to an R1 probability.
     
   • We then associate the probability P with this number.
     
   • Every probability can be associated with a real number, rational or irrational.
     
   • We still have to prove that the results given by our rules are consistent; that is, if a probability P is greater than another probability Q, that the number associated with P by our rules is greater than that associated with Q.
     
   • The probability-number that one of t of these sets contains the true proposition is t/m.
     
   • But this is also the probability-number that one of ts propositions selected from the original ms propositions shall be the true one, which by our rule is ts/ms and equal to t/m, as it should be.
     
   • If then P is greater than Q, the number of alternatives needed to give probability P must exceed that needed to give probability Q; therefore ts is greater than rm.
     
   • But this is equivalent to saying that t/m is greater than r/s; and therefore the greater probability is associated with the greater number.
     
   • Then the probability associated with a is greater than that associated with t/m, and that associated with t/m is greater than that associated with b.
     
   • Hence, in virtue of the transitive property of the relation more probable than, the probability associated with a is greater than that associated with b.
     
   • In other words, the greater number corresponds to the greater probability.
     
   • We have seen how definite numbers can be associated with probabilities, so that the higher number always corresponds to the higher probability.
     
   • In consequence of our fundamental assumption our rules always imply the existence of a definite probability-number.
     
   • The rules, as we stated before, are conventions and not hypotheses; for if the probability-number assigned by our rules is x, any function of x that always increases with x would satisfy the fundamental assumption.
     
   • Henceforth we shall have no need to speak of probabilities apart from their associated numbers, and when we speak of the probability of a proposition on given data we shall mean the number associated with the probability by our rules.
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Jeffreys_Probability.html .
     

2. Keynes: 'Probability' Introduction Ch I
   
   • Keynes: Probability Introduction Ch I .
     
   • Keynes worked on the theory of probability and submitted a dissertation on that topic for a fellowship at King's College, Cambridge in March 1908.
     
   • We present a version of the introductory first chapter of the book on The meaning of probability.
     
   • See Keynes Intro Ch II for the second introductory chapter where Keynes looks at Probability in relation to the theory of knowledge.
     
   • THE MEANING OF PROBABILITY .
     
   • The Theory of Probability is concerned with that part which we obtain by argument, and it treats of the different degrees in which the results so obtained are conclusive or inconclusive.
     
   • Thus for a philosophical treatment of these branches of knowledge, the study of probability is required.
     
   • To this extent, therefore, probability may be called subjective.
     
   • But in the sense important to logic, probability is not subjective.
     
   • The Theory of Probability is logical, therefore, because it is concerned with the degree of belief which it is rational to entertain in given conditions, and not merely with the actual beliefs of particular individuals, which may or may not be rational.
     
   • Let our premisses consist of any set of propositions h, and our conclusion consist of any set of propositions a, then, if a knowledge of h justifies a rational belief in a of degree a, we say that there is a probability-relation of degree a between a and h.
     
   • Writers on Probability have generally dealt with what they term the "happening " of "events." In the problems which they first studied this did not involve much departure from common usage.
     
   • But these expressions are now used in a way which is vague and ambiguous; and it will be more than a verbal improvement to discuss the truth and the probability of propositions instead of the occurrence and the probability of events [The first writer I know of to notice this was Ancillon in Doutes sur les bases du calcul des probabilites (1794): "Dire qu'un fait passe, present ou a venir est probable, c'est dire qu'une proposition est probable." The point was emphasised by Boole, Laws of Thought, pp.
     
   • It is not straining the use of words to speak of this as the relation of probability.
     
   • Students of probability in the sense which is meant by the authors of typical treatises on Wahrscheinlichkeitsrechnung or Calcul des probabilites, will find that I do eventually reach topics with which they are familiar.
     
   • As soon as mathematical probability ceases to be the merest algebra or pretends to guide our decisions, it immediately meets with problems against which its own weapons are quite powerless.
     
   • And even if we wish later on to use probability in a narrow sense, it will be well to know first what it means in the widest.
     
   • This relation is the subject-matter of the logic of probability.
     
   • A great deal of confusion and error has arisen out of a failure to take due account of this relational aspect of probability.
     
   • No proposition is in itself either probable or improbable, just as no place can be intrinsically distant; and the probability of the same statement varies with the evidence presented, which is, as it were, its origin of reference.
     
   • We may fix our attention on our own knowledge and, treating this as our origin, consider the probabilities of all other suppositions, - according to the usual practice which leads to the elliptical form of common speech; or we may, equally well, fix it on a proposed conclusion and consider what degree of probability this would derive from various sets of assumptions, which might constitute the corpus of knowledge of ourselves or others, or which are merely hypotheses.
     
   • There is nothing novel in the supposition that the probability of a theory turns upon the evidence by which it is supported; and it is common to assert that an opinion was probable on the evidence at first to hand, but on further information was untenable.
     
   • A definition of probability is not possible, unless it contents us to define degrees of the probability-relation by reference to degrees of rational belief.
     
   • We cannot analyse the probability-relation in terms of simpler ideas.
     
   • As soon as we have passed from the logic of implication and the categories of truth and falsehood to the logic of probability and the categories of knowledge, ignorance, and rational belief, we are paying attention to a new logical relation in which, although it is logical, we were not previously interested, and which cannot be explained or defined in terms of our previous notions.
     
   • I do not believe that any of them accurately represent that particular logical relation which we have in our minds when we speak of the probability of an argument.
     
   • In the case of "probability" the object before the mind is so familiar that the danger of misdescribing its qualities through lack of a definition is less than if it were a highly abstract entity far removed from the normal channels of thought.
     
   • The view, that probability arises out of the existence of a specific relation between premiss and conclusion, depends for its acceptance upon a reflective judgment on the true character of the concept.
     
   • It will be our object to discuss, under the title of Probability, the principal properties of this relation.
     

3. Keynes: 'Probability' Introduction Ch II
   
   • Keynes: Probability Introduction Ch II .
     
   • Keynes worked on the theory of probability and submitted a dissertation on that topic for a fellowship at King's College, Cambridge in March 1908.
     
   • We present a version of the introductory second chapter of the book on Probability in relation to the theory of knowledge.
     
   • See Keynes Intro Ch I for the first introductory chapter where Keynes looks at The meaning of probability.
     
   • PROBABILITY IN RELATION TO THE THEORY OF KNOWLEDGE .
     
   • If the evidence upon which we base our belief is h, then what we know, namely q, is that the proposition p bears the probability-relation of degree a to the set of propositions h; and this knowledge of ours justifies us in a rational belief of degree a in the proposition p.
     
   • It will be convenient to call propositions such as p, which do not contain assertions about probability-relations, "primary propositions"; and propositions such as q, which assert the existence of a probability-relation, "secondary propositions." [This classification of "primary" and "secondary" propositions was suggested to me by Mr W E Johnson.] .
     
   • At this point it is desirable to colligate the three senses in which the term probability has been so far employed.
     
   • In its most fundamental sense, I think, it refers to the logical relation between two sets of propositions, which in § 4 of Chapter I, I have termed the probability-relation.
     
   • Derivative from this sense, we have the sense in which, as above, the term probable is applied to the degrees of rational belief arising out of knowledge of secondary propositions which assert the existence of probability-relations in the fundamental logical sense.
     
   • Further it is often convenient, and not necessarily misleading, to apply the term probable to the proposition which is the object of the probable degree of rational belief, and which bears the probability-relation in question to the propositions comprising the evidence.
     
   • Now our knowledge of propositions seems to be obtained in two ways: directly, as the result of contemplating the objects of acquaintance; and indirectly, by argument, through perceiving the probability-relation of the proposition, about which we seek knowledge, to other propositions.
     
   • The logic of knowledge is mainly occupied with a study of the logical relations, direct acquaintance with which permits direct knowledge of the secondary proposition asserting the probability-relation, and so to indirect knowledge about, and in some cases of, the primary proposition.
     
   • Knowledge, on the other hand, of a secondary proposition involving a degree of probability lower than certainty, together with knowledge of the premiss of the secondary proposition, leads only to a rational belief of the appropriate degree in the primary proposition.
     
   • Of probability we can say no more than that it is a lower degree of rational belief than certainty; and we may say, if we like, that it deals with degrees of certainty.
     
   • The view, occasionally held, that probability is concerned with degrees of truth, arises out of a confusion between certainty and truth.
     
   • Perhaps the Aristotelian doctrine that future events are neither true nor false arose in this way.] Or we may make probability the more fundamental of the two and regard certainty as a special case of probability, as being, in fact, the maximum probability.
     
   • Thus a proposition is impossible with respect to a given premiss, if it is disproved by the premiss; and the relation of impossibility is the relation of minimum probability.
     
   • [Necessity and Impossibility, in the senses in which these terms are used in the theory of Modality, seem to correspond to the relations of Certainty and Impossibility in the theory of probability, the other modals, which comprise the intermediate degrees of possibility, corresponding to the intermediate degrees of probability.
     
   • Almost up to the end of the seventeenth century the traditional treatment of modals is, in fact, a primitive attempt to bring the relations of probability within the scope of formal logic.] .
     
   • In order that we may have rational belief in p of a lower degree of probability than certainty, it is necessary that we know a set of propositions h, and also know some secondary proposition q asserting a probability-relation between p and h.
     
   • Such belief can only arise, that is to say, by means of the perception of some probability-relation.
     
   • I assume then that only true propositions can be known, that the term "probable knowledge" ought to be replaced by the term "probable degree of rational belief," and that a probable degree of rational belief cannot arise directly but only as the result of an argument, out of the knowledge, that is to say, of a secondary proposition asserting some logical probability-relation in which the object of the belief stands to some known proposition.
     
   • The perceptions of some relations of probability may be outside the powers of some or all of us.
     
   • What we know and what probability we can attribute to our rational beliefs is, therefore, subjective in the sense of being relative to the individual.
     
   • With these brief indications as to the relation of Probability, as I understand it, to the Theory of Knowledge, I pass from problems of ultimate analysis and definition, which are not the primary subject matter of this book, to the logical theory and superstructure, which occupies an intermediate position between the ultimate problems and the applications of the theory, whether such applications take a generalised mathematical form or a concrete and particular one.
     

4. Aitken: 'Statistical Mathematics
   
   • STATISTICS AS A SCIENCE: AXIOMS OF PROBABILITY .
     
   • What is the axiomatic basis of the science of statistics, and what are the facts upon which the inductive synthesis is based? The facts are certain regularities which have been observed in the proportionate frequency with which certain simple events happen or do not happen, when the circumstances under which they may occur are reconstructed again and again in repeated trials; and the axioms, and the structure of theorems founded upon them, constitute the subject called mathematical probability.
     
   • Simple ideas such as these suggest by generalization and abstraction the axioms of probability; but the choice of axioms may be made in various ways, which lead to different formulations of the theory of probability.
     
   • Survey of Various Definitions of Probability.
     
   • No single particular definition of probability has so far met with predominating acceptance.
     
   • Probability as the Logic of Uncertain Inference.
     
   • One view is that probability may be regarded as a kind of extension of classical logic, an extension conveniently described as the "logic of uncertain inference." This view has been expounded by J M Keynes in A Treatise on Probability (London, 1921), especially in Part II, Chapters X-XVII, where references to earlier expositions are given.
     
   • Probability is here regarded as "the degree of our rational belief" in the truth of a given proposition, such belief being contingent on a body of relevant knowledge.
     
   • Probability a Priori, and Probability as Relative Frequency.
     
   • As our simple illustrations of the coin and the die have suggested, the crude intuition of probability rests on the observation that when a given set of circumstances S, such as a symmetrical coin spun rapidly, has been present on numerous occasions in the past, it has been associated in a nearly constant proportion of those occasions with some event E, such as the fall of "heads." .
     
   • However, symmetry being presumed, the six faces 1, 2, 3, 4, 5, 6 were characterized as "equally likely" to be found uppermost after any throw, and the probability of 1/6 was attributed to each of these "events." More generally, if n equally likely aspects of a proposed system S were discriminated, m of these being favourable to the event E, the probability of E with respect to S was defined as p(E ; S) = m/n.
     
   • The logician will not fail to pounce upon the words "equally likely," pointing out that they are synonymous with "equally probable", and that therefore probability is being defined by what is probable, a circulus in definiendo being thus committed.
     
   • Postponing the defence, we may pass on to inquire what could be the definition of probability, should the tests have disclosed asymmetry in S.
     
   • It follows that in order to define a probability p(E ; S) which shall be unique and not discordant with experience, we must idealize once again, postulating a limiting process as n tends to infinity and writing .
     
   • It would seem that the only kind of assertion about αn which would carry conviction would itself involve somewhere the notion of probability; and here the risk of committing a circle in definition again raises its head.
     
   • It should be added that the chief defects of the approach to probability by limit of frequency ratio have lately been removed by the work of von Mises, Copeland, Dorge, Wald and others.
     
   • Probability as Measure of a Sub-Aggregate.
     
   • If M is the measure of the whole aggregate S of possible phases, and pM the measure of the aggregate of E-phases contained in it, then p is the probability p(E ; S).
     
   • This has long been known in problems of so-called geometrical probability.
     
   • For example, given a circle, let a chord be drawn across it at random: what is the probability that the length of the chord exceeds half the diameter? It depends entirely on the manner in which the chord is drawn.
     
   • If it is done by taking a point on the circumference and then drawing the chord at any angle, all angles being thus supposed equally likely, then the probability is 2/3; but if it is done by taking any diameter and drawing the chord at right angles to any point taken in the diameter, the diameters and points being equally likely, then the probability is √3/2.
     
   • (iv) a measure M can be given to the whole set S, and if pM is the measure of the subset favourable to E, then p is the probability p(E ; S) of E with respect to S; .
     
   • Let us finally add that the word phase can be extended to include coordinates other than dynamical ones; also that the name "fundamental probability set" is used by some writers for the set S of phases Sj .
     

5. Keynes: 'Probability' Preface
   
   • Keynes: Probability Preface .
     
   • Keynes worked on the theory of probability and submitted a dissertation on that topic for a fellowship at King's College, Cambridge in March 1908.
     
   • See Keynes Intro Ch I for the first introductory chapter where Keynes looks at The meaning of probability.
     
   • See Keynes Intro Ch II for the first introductory chapter where Keynes looks at Probability in relation to the theory of knowledge.
     
   • ON PROBABILITY .
     
   • The subject matter of this book was first broached in the brain of Leibniz, who, in the dissertation, written in his twenty-third year, on the mode of electing the kings of Poland, conceived of Probability as a branch of Logic.
     
   • A few years before, "un probleme," in the words of Poisson, "propose a un austere janseniste par un homme du monde, a ete l'origine du calcul des probabilites." In the intervening centuries the algebraical exercises, in which the Chevalier de la Mere interested Pascal, have so far predominated in the learned world over the profounder enquiries of the philosopher into those processes of human faculty which, by determining reasonable preference, guide our choice, that Probability is oftener reckoned with Mathematics than with Logic.
     

6. Bertrand's work on probability' Introduction
   
   • Bertrand's work on probability Introduction .
     
   • Oscar Sheynin wrote an article Bertrand's work on probability which appeared in Arch.
     
   • A few of his notes on the theory of probability and combination of observations appeared in 1875-1884.
     
   • His last note on probability was dated 1892.
     
   • M Levy (1900) indicates that Bertrand taught probability "a diverses reprises" both at the "College dans son enseignement moins eleve" and at the Ecole Darboux (1902) testifies that in 1878 Bertrand abandoned his teaching at the College, but then, in 1886, had to resume his activities there.
     
   • This fact likely explains Bertrand's sudden interest in probability as manifested by his publications of 1887-1888.
     
   • For the first time ever, I describe in full Bertrand's work on probability and error theory.
     
   • http://www-history.mcs.st-andrews.ac.uk/Extras/Bertrand_probability.html .
     

7. R A Fisher: 'Statistical Methods' Introduction
   
   • The deduction of inferences respecting samples, from assumptions respecting the populations from which they are drawn, shows us the position in Statistics of the classical Theory of Probability.
     
   • For a given population we may calculate the probability with which any given sample will occur, and if we can solve the purely mathematical problem presented, we can, calculate the probability of occurrence of any given statistic calculated from such a sample.
     
   • The problems of distribution may in fact be regarded as applications and extensions of the theory of probability.
     
   • Three of the distributions with which we shall be concerned, Bernoulli's binomial distribution, Laplace's normal distribution, and Poisson's series, were developed by writers on probability.
     
   • For many years, extending over a century and a half, attempts were made to extend the domain of the idea of probability to the deduction of inferences respecting populations from assumptions (or observations) respecting samples.
     
   • Such inferences are usually distinguished under the heading of inverse Probability, and have at times gained wide acceptance.
     
   • This is not the place to enter into the subtleties of a prolonged controversy; it will be sufficient in this general outline of the scope of Statistical Science to reaffirm my personal conviction, which I have sustained elsewhere, that the theory of inverse probability is founded upon an error, and must be wholly rejected.
     
   • Inferences respecting populations, from which known samples have been drawn, cannot by this method be expressed in terms of probability, except in the trivial case when the population is itself a sample of a superpopulation the specification of which is known with accuracy.
     
   • The probabilities established by those tests of significance, which we shall later designate by t and z, are, however, entirely distinct from statements of inverse probability, and are free from the objections which apply to these latter.
     
   • Their interpretation as probability statements respecting populations constitute an application unknown to the classical writers on probability.
     
   • To distinguish such statements as to the probability of causes from the earlier attempts now discarded, they are known as statements of Fiducial Probability.
     
   • The rejection of the theory of inverse probability was for a time wrongly taken to imply that we cannot draw, from knowledge of a sample, inferences respecting the corresponding population.
     
   • What has now appeared is that the mathematical concept of probability is, in most cases, inadequate to express our mental confidence or diffidence in making such inferences, and that the mathematical quantity which appears to be appropriate for measuring our order of preference among different possible populations does not in fact obey the laws of probability.
     
   • To distinguish it from probability, I have used the term "Likelihood" to designate this quantity [A more special application of the likelihood is its use, under the name of "power function," for comparing the sensitiveness, in some chosen respect, of different possible tests of significance.]; since both the words "likelihood" and "probability" are loosely used in common speech to cover both kinds of relationship.
     
   • The solutions of problems of distribution (which may be regarded as purely deductive problems in the theory of probability) not only enable us to make critical tests of the significance of statistical results, and of the adequacy of the hypothetical distributions upon which our methods of numerical inference are based, but afford real guidance in the choice of appropriate statistics for purposes of estimation.
     

8. De Montmort: 'Essai d'Analyse
   
   • After his marriage he settled down on his country estate and set himself to work on the theory of probability.
     
   • The fact that he wrote at all is probably a fortunate one for the probability calculus.
     
   • This is possibly the first exponential limit in the calculus of probability, but having achieved it Montmort can't make much use of it.
     
   • The principle of conditional probability, often attributed to de Moivre but probably dating back to the controversy between Huygens and Hudde, is used with facility and understanding.
     
   • dice, in which the principles of calculating a conditional probability have been already laid down, using his cumbersome combinatorial notation, he does not appear to achieve anything else which is new.
     
   • Having reached this point he decides that while the rules of probability can be applied to the game of life, the chances of this game are too difficult to compute, much as it is too difficult to compute the value of a throw in backgammon:- .
     
   • a memoir in which it is proposed to estimate the probability of men speaking the truth, whether in speech or in writing: but can one find this? If an emphatic yes gives a semblance of truth a/b, an emphatic yes of an emphatic yes will give a semblance a/b.c/d of truth if the witness of the second is not of the same strength, and a2/b2ifthey hold the same authority, ..
     
   • The generalisations of the various topics discussed in the first edition are interesting, without adding anything particularly new to the probability calculus, although the various methods for the summation of series show the skill of the Bernoullis in that part of algebra.
     
   • [Apart from de Moivre's correspondence with Johann Bernoulli, which has nothing to do with probability, there are no letters written by de Moivre known to the present writer and a search has so far failed to find any.
     
   • With the publication of this second edition Montmort seems to have given up researches on the probability calculus.
     
   • It may have been that the short history which he wrote about the theory of probability (or possibly the calculus controversy) piqued his curiosity, but he wrote to Nicolaus (August 20th, 1713):- .
     
   • Montmort's importance from the probability point of view is possibly not in the new ideas which he introduced but in the algebraic methods of attack.
     
   • They must have given inspiration to many other pure mathematicians, among them de Moivre, who would not have been interested solely in the laborious enumeration of the fundamental probability set.
     

9. Harold Jeffreys on Logic and Scientific Inference
   
   • Between 1919 and 1923 Harold Jeffreys and Dorothy Wrinch wrote three papers on probability and scientific inference.
     
   • D Wrinch and H Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 38 (1919), 715-731.
     
   • In their views of probability they were influenced by William Ernest Johnson and John Maynard Keynes.
     
   • Men we make a scientific generalization we do not assert the generalization or its consequences with certainty; we assert that they have a high degree of probability on the knowledge available to us at the time, but that this probability may be modified by additional knowledge.
     
   • The more facts are shown to be co-ordinated by a law, the higher the probability of that law and of further inferences from it.
     
   • The notion of probability, which plays no part in logic, is fundamental in scientific inference.
     
   • We must consider what general rules it satisfies, what probabilities are attached to propositions in particular cases, and how the theory of probability can be developed so as to derive estimates of the probabilities of propositions inferred from others and not directly known by experience.
     
   • Thus the classifications of sensations actually adopted in practical description are determined by considerations derived from the theory of inference; and probability, from being a despised and generally avoided subject, becomes the most fundamental and general guiding principle of the whole of science.
     
   • See also Jeffreys Probability .
     

10. A I Khinchin: 'Statistical Mechanics' Introduction
    
    • All these requirements are satisfied best by the methods of the theory of probability.
      
    • It is clear that the well-known trends of the theory of probability fit in the best possible way the aforementioned special demands of the molecular-physical theories.
      
    • Secondly, the notions of the theory of probability do not appear in a precise form and are not free from a certain amount of confusion which often discredits the mathematical arguments by making them either void of any content or even definitely incorrect.
      
    • The limit theorems of the theory of probability do not find any application as yet.
      
    • (1) A precise introduction of the notion of probability, which is given here a purely mechanical definition, is lacking with the resulting questionable logical precision of all arguments of statistical character.
      
    • (2) The limit theorem of the theory of probability does not find any application (at that time they were not quite developed in the theory of probability itself).
      
    • To a considerable extent this abstruseness was due to the fact that the authors did not use the limit theorems of the theory of probability (sufficiently developed by that time), but created anew the necessary analytical apparatus.
      
    • In places where we might have to use finite sums or series, we operate with integrals, continuous distributions of probability might be replaced by the discrete ones, for which completely analogous limit theorems hold true.
      

11. Max Planck: 'Quantum Theory
    
    • Since at that time I did not see my way clear to go any further into the dependence of entropy and probability, I could, first of all, only refer to results that had already been obtained.
      
    • Therefore, since it was first enunciated, I have been trying to give it a real physical meaning, and this problem led me to consider the relation between entropy and probability, along the lines of Boltzmann's ideas.
      
    • According to Boltzmann, entropy is a measure of physical probability, and the essence of the second law of thermo-dynamics is that in Nature, the more often a condition occurs, the more probable it is.
      
    • As in the case of energy, we can define absolute value for entropy and consequently for physical probability, if the additive constant is fixed so that entropy and energy vanish simultaneously.
      
    • (It would be better to substitute temperature for energy here.) On this basis a comparatively simple combinatory method was derived for calculating the physical probability of a certain distribution of energy in a system of resonators.
      
    • For numerical applications of this method of probability we require two universal constants, each of which has an independent physical significance.
      
    • Though it was indispensable for obtaining the right expression for entropy for it is only by the help of it that the magnitude of the standard element of probability could be fixed for the probability calculations - it proved itself unwieldy and cumbrous in all attempts to make it fit in with classical theory in any form.
      

12. Studies presented to Richard von Mises' Introduction
    
    • He examined and presented this role in a precise and lucid way and removed the obscurity that bad been inherent in the traditional presentation of statistics and probability.
      
    • Thus a very rational line of thought connects von Mises' work in mechanical engineering (Theorie der Wasserrader, Fluglehre, etc.) with his investigations into the logical foundations of probability.
      
    • If we study his work in fields of such complex structure as plasticity or turbulence, we never find smug contentment with rules of thumb or quick transitions from a vague assumption to a long row of figures, but meet everywhere the attempt to analyze these difficult problems in terms of rational mechanics and to examine critically "die bisherigen Ansatze." We see him, on the other hand, freeing probability theory from semi-mystical formulations, according to which the concept of probability is derivable from our "ignorance." To do this, he had to construct a system of statements, based, as is every physical theory, upon the combination of a formal system and the physical interpretation of its terms.
      
    • In probability as well as in mechanical engineering, von Mises has investigated the complete range of problems that stretches from the construction of a suitable formal system to methods of numerical computation.
      
    • In Probability, Statistics, and Truth, von Mises offered a brilliant presentation of his general ideas on probability to a wider class of readers; it is perhaps still the best book to make a general scientist or, for that matter, any well-educated person familiar with the conception of probability and its applications.
      

13. Born Inaugural
    
    • The probability of being hit has dropped enormously, but if you are hit the effect is just as fatal as before.
      
    • If the latter fires not only horizontally but equally in all directions, the number of bullets and therefore the probability of being hit, will decrease with distance in exactly the same ratio as the surface of the concentric spheres, over which the bullets are equally distributed, increases.
      
    • I have generalised this idea for electrons and any other kind of particles by the statement that we have to do with "waves of probability" guiding the particles in such a way that the intensity of the wave at a point is always proportional to the probability of finding a particle at that point.
      
    • The corresponding probability wave must also be restricted to this small part of space, according to our second quantum law.
      
    • In the case of causality there also exists a more general conception, that of probability.
      
    • Necessity is a special case of probability; it is a probability of one hundred per cent.
      

14. Rota's lecture on 'Mathematical Snapshots
    
    • What is the probability that the point shall belong to the smaller set A? The answer is obvious: such a probability equals the volume of A divided by the volume of B.
      
    • Assuming that such a straight line meets the larger set B, what is the probability that such a straight line will also meet the smaller set A? .
      
    • Such a probability equals the surface area of the set A, divided by the surface area of the set B.
      
    • Assuming that the plane meets the larger set B, what is the probability that it will also meet the smaller set A? Such a probability equals the mean width of A, divided by the mean width of B.
      

15. Kelvin on the sun, Part 2
    
    • Lane remarks that the density at the centre of the sun would be "nearly one-third greater than that of the metal platinum," if the gaseous law held up to so great a degree of condensation for the ingredients of the sun's mass; but he does not suggest this supposition as probable, and he no doubt agrees with the general opinion that in all probability the ingredients of the sun's mass, at the actual temperatures corresponding to their positions in his interior, obey the simple gaseous law through but a comparatively small space inwards from the surface; and that in the central regions they are much less condensed than according to that law.
      
    • According to the simple gaseous law, the sun's central density would be thirty-one times that of water; we may assume that it is in all probability much less than this, though considerably greater than the mean density, 1.4.
      
    • Five or ten million years ago he may have been about double his present diameter and an eighth of his present mean density, or 0.175 of the density of water; but we cannot, with any probability of argument or speculation, go on continuously much beyond that.
      
    • We cannot, however, help asking the question, What was the condition of the sun's matter before it came together and became hot? It may have been two cool solid masses, which collided with the velocity due to their mutual gravitation; or, but with enormously less of probability, it may have been two masses colliding with velocities considerably greater than the velocities due to mutual gravitation.
      
    • This exceedingly exact aiming of the one body at the other, so to speak, is, on the dry theory of probability, exceedingly improbable.
      
    • Thus we see that the dry probability of collision between two neighbours of a vast number of mutually attracting bodies widely scattered through space is much greater if the bodies be all given a rest, than if they be given moving in any random directions and with any velocities considerable in comparison with the velocities which they would acquire in falling from rest into collision.
      

16. Harold Jeffreys: 'Scientific Inference' Preface
    
    • Between 1919 and 1923 Harold Jeffreys and Dorothy Wrinch wrote three papers on probability and scientific inference.
      
    • D Wrinch and H Jeffreys, On Some Aspects of the Theory of Probability, Philosophical Magazine 38 (1919), 715-731.
      
    • In their views of probability they were influenced by William Ernest Johnson and John Maynard Keynes.
      
    • See also Jeffreys Inference and Jeffreys Probability .
      
    • See also Jeffreys Inference and Jeffreys Probability .
      

17. Leonard J Savage: 'Foundations of Statistics
    
    • In physics, 'statistics' usually pertains to probability without special reference to the problem of inference but with emphasis on large aggregates.
      
    • A certain subjective theory of probability formulated by Ramsey [The Foundations of Mathematics and Other Logical Essays (Kegan Paul, London, 1931).',9)" onmouseover="window.status='Click to see reference';return true">9] and later and more extensively by de Finetti [Ann.
      
    • It gives, a few of us believe, a consistent, workable, and unifying analysis for all problems about the interpretation of the theory of probability, a much contested subject.
      
    • None the less, the economic outlook and the subjectivistic theory of probability lend strong support to the likelihood-ratio doctrine and promise to hasten its acceptance and exploitation.
      
    • Secondly, David Wallace has recently obtained a valuable new insight into the much vexed Behrens-Fisher problem by reconsidering it from the point of view of subjective probability.
      

18. A I Khinchin: 'Statistical Mechanics' Preface
    
    • In the books on physics the formulation of the fundamental notions of the theory of probability as a rule is several decades behind the present scientific level, and the analytic apparatus of the theory of probability, mainly its limit theorems, which could be used to establish rigorously the formulas of statistical mechanics without any complicated special machinery, is completely ignored.
      
    • The present book considers as its main task to make the reader familiar with the mathematical treatment of statistical mechanics on the basis of modem concepts of the theory of probability and a maximum utilization of its analytic apparatus.
      
    • The only essentially new material in this book consists in the systematic use of limit theorems of the theory of probability for rigorous proofs of asymptotic formulas without any special analytic apparatus.
      

19. EMS 1938 Colloquium
    
    • Professor M Frechet, of the university of Paris, who recently published a book on "The Definition of Probability" in two lectures expounded the diverse definitions which have been given of the probability of an event and has compared their respective values.
      
    • The arrangement of the timetable encouraged the morning coffee habit; so the cafe gardens of the town, and in lesser concentration the roads and walks about the place, were the scene of much deep talk on the foundations of probability, theories of the universe, and the personalities of mathematicians.
      
    • Professor M Frechet lectured and introduced a discussion on the various definitions of probability.
      

20. Laplace: 'Essay on probabilities
    
    • I present here, without the aid of analysis, the principles and general results of this theory, applying them to the most important questions of life which are indeed, for the most part, only problems of probability.
      
    • Probability is relative, in part to this ignorance, in part to our knowledge.
      
    • The theory of chance consists in reducing all events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favourable to the event whose probability is sought.
      
    • The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favourable cases and whose denominator is the number of all cases possible.
      

21. Max Planck and the quanta of energy
    
    • Since I was, however, at that time still too far oriented towards the phenomenological aspect to come to closer quarters with the connection between entropy and probability, I saw myself, at first, relying solely upon the existing results of experience.
      
    • For this reason, I busied myself, from then on, that is, from the day of its establishment, with the task of elucidating a true physical character for the formula, and this problem led me automatically to a consideration of the connection between entropy and probability, that is, Boltzmann's trend of ideas; until after some weeks of the most strenuous work of my life, light came into the darkness, and a new undreamed-of perspective opened up before me.
      
    • Whilst it was completely indispensable for obtaining the correct expression for entropy - since only with its help could the magnitude of the "elementary regions" or "free rooms for action" of the probability, decisive for the assigned probability consideration, be determined - it proved elusive and resistant to all efforts to fit it into the framework of classical theory.
      

22. R A Fisher: 'History of Statistics
    
    • Thomas Bayes' celebrated essay published in 1763 is well known as containing the first attempt to use the theory of probability as an instrument of inductive reasoning; that is, for arguing from the particular to the general, or from the sample to the population.
      
    • He admitted the principle of inverse probability, quite uncritically, into the foundations of his exposition.
      
    • He perceived the aptness for this purpose of the Method of Maximum Likelihood, although he attempted to derive and justify this method from the principle of inverse probability.
      
    • The method has been attacked on this ground, but it has no real connection with inverse probability.
      

23. A I Khinchin on Information Theory
    
    • Information theory is one of the youngest branches of applied probability theory; it is not yet ten years old.
      
    • One was in terms of a quantity called equivocation, and the other was in terms of the probability of error.
      
    • It is devoted to the derivation of a whole set of unrelated inequalities, each of which is a theorem of elementary probability theory (i.e., pertains only to finite spaces).
      
    • The reader acquainted with my paper The entropy concept in probability theory (Russian) (1953) will be able to begin this paper with the second chapter, returning to the first chapter only, when references to its results appear in the text.
      

24. Planck's quanta.html
    
    • Since I was, however, at that time still too far oriented towards the phenomenological aspect to come to closer quarters with the connection between entropy and probability, I saw myself, at first, relying solely upon the existing results of experience.
      
    • For this reason, I busied myself, from then on, that is, from the day of its establishment, with the task of elucidating a true physical character for the formula, and this problem led me automatically to a consideration of the connection between entropy and probability, that is, Boltzmann's trend of ideas; until after some weeks of the most strenuous work of my life, light came into the darkness, and a new undreamed-of perspective opened up before me.
      
    • Whilst it was completely indispensable for obtaining the correct expression for entropy - since only with its help could the magnitude of the "elementary regions" or "free rooms for action" of the probability, decisive for the assigned probability consideration, be determined - it proved elusive and resistant to all efforts to fit it into the framework of classical theory.
      

25. Ford - Mathematics for Field Artillery
    
    • This pamphlet comprises sections entitled "arithmetic," "algebra," "geometry," "trigonometry," "approximate methods," "coordinates," "aids to calculation," and "probability," and has also an appendix of tables.
      
    • Under the topic of probability, the probability curve and the dispersion diagram are illustrated and applied.
      

26. Eulogy to Euler by Fuss
    
    • The hypothesis was proved out by experiments and these conformed to natural laws which in turn ensured its ultimate probability.
      
    • Everything that the calculations of probability can provide is found in this important topic.
      

27. Schrödinger: 'Statistical Thermodynamics
    
    • in the way suiting the purpose in question and to count the numbers in the classes, so as to be able to judge of the probability of certain features or characteristics turning up in the assembly.
      
    • Moreover, since the N systems are alike and under similar conditions, we can then obviously, from their simultaneous statistics, judge of the probability of finding our system, when placed in a heat-bath of given temperature, in one or other of its private states.
      

28. EMS 1938 Colloquium 2.html
    
    • Professor M Frechet, of the university of Paris, who recently published a book on "The Definition of Probability" in two lectures expounded the diverse definitions which have been given of the probability of an event and has compared their respective values.
      

29. Eulogy to Euler by Fuss
    
    • The hypothesis was proved out by experiments and these conformed to natural laws which in turn ensured its ultimate probability.
      
    • Everything that the calculations of probability can provide is found in this important topic.
      

30. EMS 1938 Colloquium 4.html
    
    • The arrangement of the timetable encouraged the morning coffee habit; so the cafe gardens of the town, and in lesser concentration the roads and walks about the place, were the scene of much deep talk on the foundations of probability, theories of the universe, and the personalities of mathematicians.
      
    • Professor M Frechet lectured and introduced a discussion on the various definitions of probability.
      

31. Sheppard Papers
    
    • "Table of the probability integral." Biometrika, vol.
      
    • Various papers and lectures on probability, statistics, arithmetic, etc., to the Math.
      

32. W H Young addresses ICM 1928 Part 2
    
    • Indeed, the idea of degrading Mathematics in Biology, Sociology, Economics, Politics, to mere Statistics, [or even to the Theory of Probability, magnificent as are its promises], can only take form in the minds of men immersed in the details of purely observational work.
      
    • Thus, for instance, in the natural enquiry into the mutual relation of Complementary Colours, the attempt to discover a relation between their Wave Lengths might with fair probability have been condemned on a priori grounds: the unconcerned retention of the wave length as a working variable, after it had served only as a characterising Order-Number, and the ignoration of the second variable in the case, seem sufficiently crave errors to warrant any failure.
      

33. George Chrystal's Second Promoter's Address
    
    • As a matter of fact, the ordinances already issued, and in all probability about to become law, have gone so far that financial reconstruction cannot be further delayed without grave injury to the university.
      

34. ELOGIUM OF EULER
    
    • Probability and political arithmetic were also part of his indefatigable undertakings.
      

35. Three Sadleirian Professors
    
    • In 1928 he edited the late Professor Burnside's Theory of Probability, and in 1930 he published his own two-volume Geometry of Four Dimensions.
      

36. Charles Tweedie on James Stirling
    
    • Witness, for example, the tribute of praise rendered by Laplace in his papers on Probability and on the Laws of Functions of very large numbers.
      

37. Statistics
    
    • Bertrand Introduction to Bertrand's work on probability .
      

38. Mathematicians and Music 2.2
    
    • Discussing the probability of other planets' being inhabited and of the inhabitants' possible interest in music and invention of musical instruments, he continues:- .
      

39. Percy MacMahon addresses the British Association in 1901, Part 2
    
    • Life and form, however, were infused when it was recognised by De Moivre, Bernoulli, and others that it was possible to create a science of probability on the basis of enumeration and arrangement.
      

40. Gordon Preston on semigroups
    
    • This was my first experience of research - it was a mixture of algebra and statistics, or probability theory, and I greatly enjoyed it.
      

41. Tait's 1888 address to the graduates
    
    • And the instances I have given you form, in all probability, a mere fraction of the inconsistencies and absurdities which have found their way into this remarkable document.
      

42. Tait graduates address.html
    
    • And the instances I have given you form, in all probability, a mere fraction of the inconsistencies and absurdities which have found their way into this remarkable document.
      

43. The South-Troughton quarrel
    
    • (Drinkwater Bethune wrote lives of Galileo and Kepler in the Library of Useful Knowledge, and with Sir John Lubbock a little book On Probability, in the same series.) Maule at once insisted that Troughton & Simms should be allowed to finish their work according to the plan proposed by Sheepshanks, but only to be paid for if successful.
      

44. George Chrystal's First Promoter's Address
    
    • What the commission will, in all probability, do, - what they certainly ought to do, - is to put elasticity and, if need be, joints into the cast-iron framework of our University Constitution, which will enable us gradually, as men and money can be found, to adapt ourselves to the existing want of our time.
      

45. Kelvin on the sun
    
    • Helmholtz's form of the meteoric theory of the origin of the sun's heat, may be accepted as having the highest degree of scientific probability that can be assigned to any assumption regarding actions of prehistoric times.
      

46. A CONTRIBUTION TO THE MATHEMATICAL THEORY OF BIG GAME HUNTING
    
    • At any given moment there is a positive probability that .
      

47. Heath: Everyman's Library 'Euclid' Introduction
    
    • Thus "Axiom 11," that all right angles are equal to one another, was Euclid's "Postulate 4," while "Axiom 12" (the well-known parallel-axiom) was "Postulate 5." Further, of the first ten "Axioms" only five can, with any probability, be attributed to Euclid himself (1, 2, 3, 8, and 9 in Todhunter's edition).
      

48. James Jeans addresses the British Association in 1934
    
    • The electron did not move continuously through space and time, but jumped, and its jumps were not governed by the laws of mechanics, but to all appearance, as Einstein showed more fully four years later, by the laws of probability.
      

49. George Chrystal's Third Promoter's Address
    
    • Now I am giving this address in all probability for the last time.
      

50. Cochran: 'Sampling Techniques' Preface
    
    • Readers with advanced training in probability may find the arguments by which theorems are established rather pedestrian.
      

51. Wave versus matrix mechanics
    
    • Heisenberg invented these concepts by focusing attention on a set of quantised probability amplitudes.
      

52. Euler Elogium.html.html
    
    • Probability and political arithmetic were also part of his indefatigable undertakings.
      

53. Kuratowski: 'Set Theory and Topology' Foreword
    
    • The purpose of the present volume is to give an accessible presentation of the fundamental concepts of set theory and topology; special emphasis being placed on presenting the material from the viewpoint of its applicability to analysis, geometry, and other branches of mathematics such as probability theory and algebra.
      

54. Turnbull lectures on Colin Maclaurin
    
    • The subjects ranged from Euclid and elementary algebra to conics, fluxions, probability and Newton's Principia.
      

55. Selected papers of Edward Marczewski' Preface
    
    • Until the late fifties his main fields of interest were measure theory, descriptive set theory, general topology and probability theory.
      

56. NAS Award in Mathematics
    
    • for the theory of free probability, in particular, using random matrices and a new concept of entropy to solve several hitherto intractable problems in von Neumann algebras.
      

57. NAS Award in Applied Mathematics and Numerical Analysis
    
    • for his brilliant and productive mathematical work encompassing genetics, economics, approximation theory, probability, and statistics, and game theory.
      

Quotations

1. Quotations by Keynes
   
   • It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability.
     
   • Probability begins and ends with probability.
     
   • The Application of Probability to Conduct.
     
   • It has been pointed out already that no knowledge of probabilities, less in degree than certainty, helps us to know what conclusions are true, and that there is no direct relation between the truth of a proposition and its probability.
     
   • Probability begins and ends with probability.
     
   • The Application of Probability to Conduct .
     

2. Quotations by Feller
   
   • Probability is a mathematical discipline whose aims are akin to those, for example, of geometry of analytical mechanics.
     
   • An Introduction to Probability Theory and its Applications .
     
   • All possible definitions of probability fall short of the actual practice.
     
   • An Introduction to Probability Theory and its Applications .
     

3. A quotation by Kolmogorov
   
   • The theory of probability as mathematical discipline can and should be developed from axioms in exactly the same way as Geometry and Algebra.
     
   • Foundations of the Theory of Probability .
     

4. Quotations by Kac
   
   • Steinhaus, with his predilection for metaphors, used to quote a Polish proverb, 'Forturny kolem sie tocza' [Luck runs in circles], to explain why π, so intimately connected with circles, keeps cropping up in probability theory and statistics, the two disciplines which deal with randomness and luck.
     
   • Statistical Independence in Probability Analysis and Number Theory .
     

5. Quotations by Littlewood
   
   • We come finally, however, to the relation of the ideal theory to real world, or "real" probability.
     

6. A quotation by Woodward
   
   • Probability and Theory of Errors .
     

7. Quotations by Peirce Charles
   
   • This branch of mathematics [Probability] is the only one, I believe, in which good writers frequently get results which are entirely erroneous.
     

8. Quotations by Pascal
   
   • The excitement that a gambler feels when making a bet is equal to the amount he might win times the probability of winning it.
     

9. A quotation by Mises
   
   • Probability, Statistics, and Truth .
     

10. Quotations by Borel
    
    • Probability and Certainty .
      

11. Quotations by Boole
    
    • Probability is expectation founded upon partial knowledge.
      

12. Quotations by Aristotle
    
    • Such an event is probable in Agathon's sense of the word: 'it is probable,' he says, 'that many things should happen contrary to probability.' .
      

Chronology

1. Mathematical Chronology
   
   • Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they exchange during the summer.
     
   • It is the first published work on probability theory, outlining for the first time the concept called mathematical expectation based on the ideas in the letters of Fermat and Pascal from 1654.
     
   • This paper is the first application of probability to social statistics.
     
   • Jacob Bernoulli's book Ars conjectandi (The Art of Conjecture) is an important work on probability.
     
   • Much of this probability treatise is based on the work of de Moivre.
     
   • Bayes publishes An Essay Towards Solving a Problem in the Doctrine of Chances which gives Bayes theory of probability.
     
   • Buffon carries out his probability experiment calculating π by throwing sticks over his shoulder onto a tiled floor and counting the number of times the sticks fell across the lines between the tiles.
     
   • Condorcet publishes Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix (Essay on the Application of the Analysis to the Probability of Majority Decisions).
     
   • It is a major advance in the study of probability in the social sciences.
     
   • Condorcet publishes Essay on the Application of Analysis to the Probability of Majority Decisions which is an extremely important work in the development of the theory of probability.
     
   • The first book studies generating functions and also approximations to various expressions occurring in probability theory.
     
   • The second book contains Laplace's definition of probability, Bayes's rule, and mathematical expectation.
     
   • In this work he establishes the rules of probability, gives "Poisson's law of large numbers" and describes the "Poisson distribution" for a discrete random variable which is a limiting case of the binomial distribution.
     
   • Keynes publishes his Treatise on Probability in which he argues that probability is a logical relation and so it is objective.
     
   • A statement involving probability relations has a truth-value independent of people's opinions.
     
   • Von Mises publishes Probability, Statistics and Truth.
     
   • Von Mises introduces the idea of a sample space into probability theory.
     
   • Kolmogorov publishes Foundations of the Theory of Probability which presents an axiomatic treatment of probability.
     
   • Kolmogorov publishes Analytic Methods in Probability Theory which lays the foundations of the theory of Markov random processes.
     
   • Carnap publishes Logical Foundations of Probability.
     

2. Chronology for 1780 to 1800
   
   • Condorcet publishes Essai sur l'application de l'analyse a la probabilite des decisions rendues a la pluralite des voix (Essay on the Application of the Analysis to the Probability of Majority Decisions).
     
   • It is a major advance in the study of probability in the social sciences.
     
   • Condorcet publishes Essay on the Application of Analysis to the Probability of Majority Decisions which is an extremely important work in the development of the theory of probability.
     

3. Chronology for 1920 to 1930
   
   • Keynes publishes his Treatise on Probability in which he argues that probability is a logical relation and so it is objective.
     
   • A statement involving probability relations has a truth-value independent of people's opinions.
     
   • Von Mises publishes Probability, Statistics and Truth.
     

4. Chronology for 1930 to 1940
   
   • Von Mises introduces the idea of a sample space into probability theory.
     
   • Kolmogorov publishes Foundations of the Theory of Probability which presents an axiomatic treatment of probability.
     
   • Kolmogorov publishes Analytic Methods in Probability Theory which lays the foundations of the theory of Markov random processes.
     

5. Chronology for 1810 to 1820
   
   • The first book studies generating functions and also approximations to various expressions occurring in probability theory.
     
   • The second book contains Laplace's definition of probability, Bayes's rule, and mathematical expectation.
     

6. Chronology for 1760 to 1780
   
   • Bayes publishes An Essay Towards Solving a Problem in the Doctrine of Chances which gives Bayes theory of probability.
     
   • Buffon carries out his probability experiment calculating π by throwing sticks over his shoulder onto a tiled floor and counting the number of times the sticks fell across the lines between the tiles.
     

7. Chronology for 1700 to 1720
   
   • This paper is the first application of probability to social statistics.
     
   • Jacob Bernoulli's book Ars conjectandi (The Art of Conjecture) is an important work on probability.
     

8. Chronology for 1650 to 1675
   
   • Fermat and Pascal begin to work out the laws that govern chance and probability in five letters which they exchange during the summer.
     
   • It is the first published work on probability theory, outlining for the first time the concept called mathematical expectation based on the ideas in the letters of Fermat and Pascal from 1654.
     

9. Chronology for 1950 to 1960
   
   • Carnap publishes Logical Foundations of Probability.
     

10. Chronology for 1940 to 1950
    
    • Carnap publishes Logical Foundations of Probability.
      

11. Chronology for 1740 to 1760
    
    • Much of this probability treatise is based on the work of de Moivre.
      

12. Chronology for 1720 to 1740
    
    • Much of this probability treatise is based on the work of de Moivre.
      

13. Chronology for 1830 to 1840
    
    • In this work he establishes the rules of probability, gives "Poisson's law of large numbers" and describes the "Poisson distribution" for a discrete random variable which is a limiting case of the binomial distribution.