Read e-book online Seminaire De Probabilites XVIII 1982/83 PDF

By Jacques Azema

ISBN-10: 0387133321

ISBN-13: 9780387133324

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86–88. 4 Pascal’s Arithmetic Triangle, taken from the Oeuvres Completes Vol. 2 (Pascal, 1858, p. 416). Edwards (2003, pp. 40–52) further adds In section 3 [of the Traite du Triangle Arithmetique]{ Pascal breaks new ground, and this section, taken together with his correspondence with Fermat, is the basis of his reputation as the father of probability theory. The correspondence between Pascal and Fermat clearly shows the latter as the consummate mathematician very much at ease with the calculus of probabilities.

Likewise he changes the number of favorable combinations for B and C, finally reaching a division ratio of 16:51/2:51/2. But he correctly notes that the answer cannot be right, for his own recursive method gives the correct ratio of 17:5:5. Thus, Pascal at first wrongly believed Fermat’s method of enumeration could not be generalized to more than two players. Fermat was quick to point out the error in Pascal’s reasoning. In his September 25, 1664 letter, Fermat explains (Smith, 1929, p. 562) In taking the example of the three gamblers of whom the first lacks one point, and each of the others lack two, which is the case in which you oppose, I find here only 17 combinations for the first and 5 for each of the others; for when you say that the combination acc is good for the first, recollect that everything that is done after one of the players has won is worth nothing.

We have E2  1, EÀ2  0, and E1 ¼ að1Þ þ bE0 ¼ a þ bE0 ; EÀ1 ¼ aE0 þ bð0Þ ¼ aE0 ; E0 ¼ aE1 þ bEÀ1 : { Although Pascal was the first to have actually used it, as we noted in Problem 4. 4 Extract from Huygens’ proof of the formula for the Gambler’s Ruin Problem, taken from the Oeuvres Completes, Vol. 14 (Huygens, 1920). { Eliminating EÀ1 and E1 , we have E0 ¼ a2 a2 : þ b2 This implies that B’s probability of ruining A is b2 =ða2 þ b2 Þ, giving a relative probability for A and B of a2 =b2 . Huygens then considers the case when both players start with four and eight units of money each, and obtains probability ratios of a4 =b4 and a8 =b8 , respectively (see Fig.

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Seminaire De Probabilites XVIII 1982/83 by Jacques Azema


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