1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

however, you can lose eternity. In the mathematical sense, Pascal’sreasoning is vague; perhaps he had no time to edit his fragment. Its meaningis, however, clear: if God exists with a fixed and however low probability,the expectation of the benefit accrued by believing in Him is infinite. Pascaldied in 1662 and the same year Arnauld & Nicole (1662/1992, p. 334)published a similar statement:Infinite things, like eternity and salvation, can not be equated to anytemporal advantage. […] We should never balance them with anythingwordly. […] The least degree of possibility of saving oneself is morevaluable than all the earthly blessings taken together, and the least peril oflosing that possibility is more considerable than all the temporal evils […].2.2.2. Huygens. Huygens was the author of the first treatise onprobability (1657). Being acquainted only with the general contents of thePascal – Fermat correspondence, he independently introduced the notion ofexpected random winning and, like those scholars, selected it as the test forsolving stochastic problems. He went on to prove that the value ofexpectation of a gambler who gets a in p cases and b in q cases was(1)pa + qbp + q.Jakob Bernoulli (1713/1999, p. 9) justified the expression (1) muchsimpler than Huygens did: if each of the p gamblers gets a, and each of the qothers receives b, and the gains of all of them are the same, then theexpectation of each is equal to (1). After Bernoulli, however, expectationbegan to be introduced formally: expressions of the type of (1) followed bydefinition.Huygens solved the problem of points under various initial conditions andlisted five additional problems two of which were due to Fermat, and one, toPascal. He solved them later, either in his correspondence, or in manuscriptspublished posthumously. They demanded the use of the addition andmultiplication theorems, the introduction of conditional probabilities and theformula (in modern notation)P(B) = ΣP(A i )P(B/A i ), i = 1, 2, …, n.Problem No. 4 was about sampling without replacement. An urncontained 8 black balls and 4 white ones and it was required to determinethe ratio of chances that in a sample of 7 balls 3 were, or were not white.Huygens determined the expectation of the former event by means of apartial difference equation (Hald 1990, p. 76). Nowadays such problemsleading to the hypergeometric distribution (J. Bernoulli 1713/1999, pp. 167– 168; De Moivre 1712/1984, Problem 14 and 1718/1756, Problem 20)appear in connection with statistical inspection of mass production.Pascal’s Problem No. 5 was the first to discuss the gambler’s ruin.Gamblers A and B undertake to score 14 and 11 points respectively in athrow of 3 dice. They have 12 counters each and it is required to determine22

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