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

3. Jakob Bernoulli and the Law **of** Large NumbersI consider Bernoulli’s main work, the Ars conjectandi (AC), which blazeda new trail by proving that statistical probability can be considered on a parwith the theoretical probability. Also described is the work **of** hiscontemporaries.Key words: law **of** large numbers, statistical probability, moral certainty,stochastic arguments3.1. Bernoulli’s Works3.1.1. The Diary (Meditationes). There, Bernoulli studied games **of**chance and the stochastic side **of** civil law. He (1975, p. 47) noted that theprobability **of** a visitation **of** a plague in a given year was equal to the ratio**of** the number **of** these visitations during a long period **of** time to the number**of** years in that period. He thus applied the definition **of** probability **of** anevent (**of** statistical probability!) rather than making use **of** chances. On p.46, in a marginal note, he wrote out the imprint **of** a review **of** Graunt’s book(§ 2.1.4) which Bernoulli possibly had not seen. But the most important inthe Meditationes is a (fragmentary) pro**of** **of** the LLN which means thatBernoulli proved it not later than in 1690.3.1.2. The Art **of** Conjecturing (1713). Its Contents. Niklaus Bernoullicompiled a Preface (J. Bernoulli 1975, p. 108) where, for the first time ever,the term calculus **of** probability (in Latin) had appeared. The book itselfcontained four parts. Interesting problems are solved in parts 1 and 3 **of** theAC (the study **of** random sums for the uniform and the binomialdistributions, a similar investigation **of** the sum **of** a random number **of**terms for a particular discrete distribution, a derivation **of** the distribution **of**the first order statistic for the discrete uniform distribution and thecalculation **of** probabilities appearing in sampling without replacement). Theauthor’s analytical methods included combinatorial analysis and calculation**of** expectations **of** winning in each set **of** finite and infinite games and theirsubsequent summing.Part 1 is a reprint **of** Huygens’ tract (§ 2.2.2). Bernoulli also compiled atable which enabled him to calculate the coefficients **of** x m in thedevelopment **of** (x + x 2 + … + x 6 ) n for small values **of** n. That polynomial tothe power **of** n was the generating function **of** the binomial (p + qx) with p =q.Part 2 dealt with combinatorial analysis and introduced the Bernoullinumbers.Part 4 contained the LLN. There also is an informal classical definition **of**probability (which Bernoulli had not applied when formulating that law), areasoning on the aims **of** the art **of** conjecturing (determination **of**probabilities for choosing the best solutions **of** problems, apparently in civillife) and elements **of** stochastic logic.Bernoulli likely considered the art **of** conjecturing as a mathematicaldiscipline based on probability as a measure **of** certainty and on expectationand including (the not yet formally introduced) addition and multiplicationtheorems and crowned by the LLN.In a letter **of** 3 Oct. 1703 Bernoulli (Kohli 1975b, p. 509) informedLeibniz about the progress in his work. He had been compiling it for many27