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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 **of**losing 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 **of**expected random winning and, like those scholars, selected it as the test forsolving stochastic problems. He went on to prove that the value **of**expectation **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