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PROBABILITY THEORY: a probable chronology Mathematical Analysis for Computer Science Lecture 1 Definition? • American Heritage Dictionary defines the Probability Theory as the branch of Mathematics that studies the likelihood of occurrence of random events in order to predict the behavior of defined systems. • So, Probability Theory, being a branch of Mathematics, is an exact, deductive science that studies uncertain quantities related to random events. • A strange marriage of mathematical certainty and uncertainty of randomness? Origin of Probability Theory • People started to use the principles of probability many years ago. • The elements of probability were applied for census of population in the ancient countries such as China, India and Egypt. • The same methods were used for estimation of the overall strength of enemy army. • Theory of probabilities as a real science came into existence in the middle of 17th century in France. French Society in the 1650’s • The time of royals and musketry, beautiful ladies and noble cavaliers! • Gambling was popular and fashionable, not restricted by law • As the games became more complicated and the stakes became larger there was a need for mathematical methods for computing chances. Enter the Mathematician • In 1654 French courtiers Antoine Gombaud, Chevalier de Méré (1607-1648) consulted Blaise Pascal (16231662) in Paris about an apparent contradiction concerning a popular dice game. • He posed some questions (known now as de Méré's problems ): – How many throws of two dice are required for a number of double six appear events will be more than a half of total throws? – problème des partis (problem of points): How to share the wagered money between two gamblers if the game interrupted untimely? Principles of Probability Theory formulated • Pascal began to correspond with his friend Pierre de Fermat (16011665) about these problems. • The correspondence between Pascal and Fermat is the origin of the mathematical study of probability. • Although a few special problems on games of chance had been solved by some Italian mathematicians in the 15th and 16th centuries, no general theory was developed before this famous correspondence. Classical Probability • The method they developed is now called the classical approach to computing probabilities. • The method: Suppose a game has n equally likely outcomes, of which m outcomes correspond to winning. Then the probability of winning is m / n. • The classical method requires a game to be broken down into equally likely outcomes. – It is not always possible to do this. – It is not always clear when possibilities are equally likely. Experience • Another method, known as the frequency method had also been used for some time. • This method consists of repeating a game a large number of times under the same conditions. The probability of winning is then approximately equal to the proportion of wins in the repeats. • This method was used by Pascal and Fermat to verify results obtained by the classical method. First Book • The Dutch scientist Christian Huygens (1629-1695), a teacher of Leibniz (1646-1716), learned of this correspondence and shortly thereafter (in 1657) published the first book on probability; entitled De Ratiociniis in Ludo Aleae. • It was a treatise on problems associated with gambling. • Huygens tract remained the only text on probability for 50 years. Early Generalizations • Jacob Bernoulli (1654-1705) proved that the frequency method and the classical method are consistent with one another in his book Ars Conjectandi in 1713. • Abraham De Moivre (1667-1754) provided many tools to make the classical method more useful, including the multiplication rule, in his book The Doctrine of Chances in 1718. • The book was popular, eventually going through three editions. From Games to Science • Throughout the 18th century, the application of probability moved from games of chance to scientific problems: – Mathematical theory of life insurance -life tables. – Biological problems -what is the probability of being born female or male? Applied Probability • Pierre-Simon Laplace (1749-1827) presented a mathematical theory of probability with an emphasis on scientific applications in his 1812 book Théorie Analytique des Probabilités. – Before Laplace, probability theory was solely concerned with developing a mathematical analysis of games of chance. – Laplace applied probabilistic ideas to many scientific and practical problems. • The theory of errors, actuarial mathematics, and statistical mechanics are examples of some of the important applications of probability theory developed in the l9th century. Probability Applied • Development of probability theory has been stimulated by the variety of its applications. • Conversely, each advance in the theory has enlarged the scope of its influence. • Mathematical statistics is one important branch of applied probability; other applications occur in such widely different fields as genetics, psychology, economics, and engineering. • Many workers have contributed to the theory since Laplace's time; among the most important are Chebyshev, Markov, von Mises, and Kolmogorov. Stagnation and Frustration • After the publication of Laplace’s book, the mathematical development of probability stagnated for many years. • By 1850, many mathematicians found the classical method to be unrealistic for general use and were attempting to redefine probability in terms of the frequency method. • These attempts were never fully accepted and the stagnation continued. Axiomatic Development • One of the difficulties is to – arrive at a definition of probability that is precise enough for use in mathematics – comprehensive enough to be applicable to a wide range of phenomena. • The search for a widely acceptable definition took nearly three centuries and was marked by much controversy. • The matter was finally resolved in the 20th century by treating probability theory on an axiomatic basis. • In 1933 a monograph Grundbegriffe der Wahrscheinlichkeitsrechnun (Foundations of the Calculus of Probabilities) by Russian mathematician A. Kolmogorov outlined an axiomatic approach that forms the basis for the modern theory. • He built up probability theory from fundamental axioms in a way comparable with Euclid's treatment of geometry. Let’s play a Game of Chance