The first book on probability theory, De Ratiociniis in Ludo Aleae , a treatise on problems of games of chance, was published by Huyghens in 1657. There were no published writings on this subject before 1657, although evidence exists that a number of fifteenth- and sixteenth-century Italian mathematicians worked out the solutions to various probability problems concerning games of chance. General methods of attack on such problems seem first to have been given by Pascal and Fermat in a celebrated correspondence, beginning in 1654. It is a fascinating cultural puzzle that the calculus of probability did not emerge until the seventeenth century, although random phenomena, such as those arising in games of chance, have always been present in man’s environment. For some enlightening remarks on this puzzle see M. G. Kendall, Biometrika , Vol. 43 (1956), pp. 9–12. A complete history of the development of probability theory during the period 1575 to 1825 is given by I. Todhunter, A History of the Mathematical Theory of Probability from the Time of Pascal to Laplace , originally published in 1865 and reprinted in 1949 by Chelsea, New York.

The work of Laplace marks a natural division in the history of probability, since in his great treatise Théorie Analytique des Probabilités , first published in 1812, he summed up his own extensive work and that of his predecessors. Laplace also wrote a popular exposition for the educated general public, which is available in English translation as A Philosophical Essay on Probabilities (with an introduction by E. T. Bell, Dover, New York, 1951).

The breadth of probability theory is today too immense for any one man to be able to sum it up. One can list only the main references in English of which the student should be aware. ¹ The literature of probability theory divides into three broad categories: (i) the nature (or foundations) of probability, (ii) mathematical probability theory, and (iii) applied probability theory.

The nature of probability theory is a subject about which competent men differ. There are at least two main classes of concepts that historically have passed under the name of “probability”. It has been suggested that one distinguish between these two concepts by calling one probability \({ }_{1}\) and the other probability \({ }_{2}\) (this terminology is suggested by R. Carnap, \(_{2}\) Logical Foundations of Probability , University of Chicago Press, 1950). The theory of probability is concerned with the problem of inductive inference, with the nature of scientific proof, with the credibility of propositions given empirical evidence, and in general with ways of reasoning from empirical data to conclusions about future experiences. The theory of probability is concerned with the study of repetitive events that appear \(_{2}\) to possess the property that their relative frequency of occurrence in a large number of trials has a stable limit value. Enlightening discussions of the theories of probability and \(_{1}\) probability \({ }_{2}\) are given, respectively, by Sir Harold Jeffreys, Scientific Inference , Second Edition, Cambridge University Press, Cambridge, 1957, and Richard von Mises, Probability, Statistics, and Truth , Second Edition, Macmillan, New York, 1957. The viewpoint of professional philosophers in regard to the nature of probability theory is debated in “A Symposium on Probability”, Philosophy and Phenomenological Research , Vol. 5 (1945), pp. 449–532, Vol. 6 (1946), pp. 11–86 and pp. 590–622. The philosophical implications of the use of probability theory in scientific explanation are examined from the point of view of the physicist in two books written for the educated layman: Max Born, Natural Philosophy of Cause and Chance , Oxford University Press, 1949, and David Bohm, Causality and Chance in Modern Physics , London, Routledge and Kegan Paul, 1957.

The mathematical theory of probability may be defined as consisting of those writings in which the viewpoint is the axiomatic one formulated in this chapter. This viewpoint developed in the twentieth century at the hands of such great probabilists as E. Borel, H. Steinhaus, P. Lévy, and A. Kolmogorov. ² The first systematic presentation of probability theory on an axiomatic basis was made in 1933 by Kolmogorov in a monograph available in English translation as Foundations of the Theory of Probability , Chelsea, New York, 1950. Several comprehensive treatises, in which are summarized the development of mathematical probability theory up to, say, 1950, are available: J. L. Doob, Stochastic Processes , Wiley, New York, 1953; B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (translated by K. L. Chung), Addison-Wesley, Cambridge, 1954; and M. Loève, Probability Theory: Foundations, Random Sequences , Van Nostrand, New York, 1955. A number of monographs covering the developments of the last twenty years are in process of preparation by various authors. The reader may gain some idea of the scope of recent work in the mathematical theory of probability by consulting the section “Probability” in the monthly publication Mathematical Reviews , which abstracts all published material on probability theory.

Applied probability theory may be defined as consisting of those writings in which probability theory enters as a tool in a scientific or scholarly investigation. There are so many fields of engineering and the physical, natural, and social sciences to which probability theory has been applied that it is not possible to cite a short list of representative references. A number of references are given in this book in the examples in which we discuss various applications of probability theory. Some idea of the diverse applications of probability theory can be gained by consulting M. S. Bartlett, Stochastic Processes , Cambridge University Press, 1955, or the book by Feller cited below. The role of probability theory in mathematical statistics is discussed in H. Cramér, Mathematical Methods of Statistics , Princeton University Press, 1946. The following books are classic introductions to probability theory that the reader can consult for alternate treatments of some of the topics discussed in this book: W. Feller, An Introduction to Probability Theory and its Applications , Second Edition, Wiley, New York, 1957; T. C. Fry, Probability and its Engineering Uses , Van Nostrand, New York, 1928; J. V. Uspensky, Introduction to Mathematical Probability , McGraw-Hill, New York, 1937. Feller’s inimitable book is especially recommended, since it is simultaneously an introductory textbook and a treatise on mathematical and applied probability theory.