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

The very fact described by formulas (1) and (2) was, however, extremelyimportant for the development **of** probability and statistics; and, anyway,should we deny the importance **of** existence theorems? Bernoulli’s resultproved that, given a large number **of** observations, statistical probabilityprovided moral certainty and was therefore not worse than the theoreticalprobability. His main aim was to discover whether the limit (2) existed andwhether it was indeed unity rather than a lesser positive number. The latterwould have meant that induction (from the ν trials) was inferior todeduction.Stochastic reasoning was now justified beyond the province **of** games **of**chance, at least for the Bernoulli trials. Strangely enough, statisticians for along time had not recognized this fact. Haush**of**er (1872, pp. 107 – 108)declared that statistics, since it was based on induction, had no intrinsicconnections with mathematics based on deduction. And Maciejewski (1911,p. 96) introduced a statistical law **of** large numbers instead **of** the Bernoulliproposition that allegedly impeded the development **of** statistics. His ownlaw qualitatively asserted that statistical indicators exhibited ever lesserfluctuations as the number **of** observations increased and his opinion likelyrepresented the prevailing attitude **of** statisticians. Bortkiewicz (1917, pp. 56– 57) thought that the LLN ought to denote a quite general fact,unconnected with any stochastic pattern, **of** a degree **of** stability **of** statisticalindicators under constant or slightly changing conditions and a large number**of** trials. Even Romanovsky (1912, p. 22; 1924, pt 1, p. 15; 1961, p. 127)kept to a similar view.That elementary understanding **of** the LLN has its prehistory, see thestatements **of** De Witt (§ 2.1.3) and Halley (§ 2.1.4). Again, it was thoughtthat the number **of** successes in n Bernoulli trials with probability p wasapproximately equal to np. Cardano applied this formula in calculationsconnected with games **of** dice (Ore 1953/1963, pp. 152 – 154 and 196).In astronomy, the arithmetic mean became the universal estimator **of** theconstant sought (§ 1.7). Recall also (§ 2.1.3) the practice **of** buying annuitiesupon several young lives. Boscovich (1758, § 481) had somewhat vaguelymaintained that the sum (not the mean!) **of** random magnitudes decreasedwith an increase in the number **of** terms (Gower 1993, p. 272). Mycorrection also applies to the other statements above to which I am nowadding Kepler (**Sheynin** 1973c, p. 120). He remarked that the total weight **of**a large number **of** metal money **of** the same coinage did not depend on theinaccuracy in the weight **of** the separate coins. Even Helmert (1905/1993, p.200) had to refute that mistake.3.2.4. Randomness and Necessity. Apparently not wishing to encroachupon theology, Bernoulli (beginning **of** Chapter 1) refused to discuss thenotion **of** randomness and subjectively described the contingent but at thebeginning **of** Chapter 4 explained randomness by the action **of** numerouscomplicated causes, cf. § 11.3. The last lines **of** his book stated that somekind **of** necessity was present even in random things. He referred to Platowho had indeed taught that after a countless number **of** centuries everythingwill return to its initial state at the moment **of** creation. In accordance withthat archaic notion **of** the Great Year, Bernoulli thus unjustifiablygeneralized the boundaries **of** his law.30