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Ars conjectandi, opus posthumum. Accedit - BERNOULLI, Jacob - 1713. [1273247]
Impensis Thurnisiorum Fratum, Basel 1713 - First edition, an exceptionally fine copy, rare in this condition. ?Jakob 1 Bernoulli?s posthumous treatise, edited by his nephew [Nicholas I Bernoulli], (the title literally means ?the art of[dice] throwing?) was the first significant book on probability theory: it set forth the fundamental principles of the calculus of probabilities and contained the first suggestion that the theory could extend beyond the boundaries of mathematics to apply to civic, moral and economic affairs. The work is divided into four parts, the first a commentary on Huygens?s De ratiociniis in ludo aleae (1657), the second a treatise on permutations (a term Bernoulli invented) and combinations, containing the Bernoulli numbers,and the third an application of the theory of combinations to various games of chance. The fourth and most important part contains Bernoulli?s philosophical thoughts on probability: probability as a measurable degree of certainty, necessity and chance, moral versus mathematical expectation, a priori and a posteriori probability, etc. It also contains his attempt to prove what is still called Bernoulli?s Theorem: that if the number of trials is made large enough, then the probability that the result will lie between certain limits will be as great as desired? (Norman). This was the first statement of the law of large numbers.#10086; PMM 179; Dibner 110; Evans 8; Grolier/Horblit 12; Sparrow 21; Norman 216.?In the first Part (pp. 2-71) Jakob Bernoulli complemented his reprint of Huygens?s tract by extensive annotations which contained important modifications and generalisations. Bernoulli?s additions to Huygens?s tract are about four times as long as the original text. The central concept in Huygens?s tract is expectation. The expectation of a player A engaged in a game of chance in a certain situation is identified by Huygens with his share of the stakes if the game is not played or not continued in a ?just? game. For the determination of expectation Huygens had given three propositions which constitute the ?theory? of his calculus of games of chance. Huygens?s central proposition III maintains:?If the number of cases I have for gaining a is p, and if the number of cases I have for gaining b is q, then assuming that all cases can happen equally easily, my expectation is worth (pa + qb)/(p + q).??Bernoulli not only gives a new proof for this proposition but also generalizes it in several ways ??Huygens?s propositions IV to VII treat the problem of points, also called the problem of the division of stakes, for two players; propositions VIII and IX treat three and more players. Bernoulli returns to these problems in Part II of the Ars Conjectandi. In his annotations to Huygens?s proposition IV he generalised Huygens?s concept of expectation ? This is the only instance in the annotations and commentaries to Huygens?s tract where Bernoulli uses the word ?probabilitas?, or probability as understood in everyday life. Later in Part IV of the Ars Conjectandi Bernoulli replaced Huygens?s main concept, expectation, by the concept of probability for which he introduced the classical measure of favourable to all possible cases. The remaining propositions X to XIV of Huygens?s tract deal with dicing problems of the kind: What are the odds to throw a given number of points with two or three dice? or: With how many throws of a die can one undertake it to throw a six or a double six? ? The meaning of Huygens?s result of proposition X, that the expectation of a player who contends to throw a six with four throws of a die is greater than that of his adversary, is explained by Bernoulli in a way which relates to the law of large numbers proved in Part IV of the Ars Conjectandi ??In the second Part (pp. 72-137) Bernoulli deals with combinatorial analysis, based on contributions of van Schooten, Leibniz, Wallis, and Jean Prestet ? [It] consists of nine chapters dealing with permutations, the number of combinations of all classes, the number of combinat [Attributes: First Edition]
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