Law of Large Numbers

LAW OF LARGE NUMBERS, in the mathematical sense (in the sense of probability theory), denotes the following proposition: with a probability close to certainty, one can expect that in a large series of experiments, in which various types of the event under consideration (or its opposite) can occur with different probabilities, the actual frequency of each of these events differs little from the arithmetic mean of the probabilities of its various types (A. K. Vlasov); in this context, frequency should be understood as the ratio of the number of cases of the realization of each of the events to the number of all experiments. The correctness of the stated proposition has been proven mathematically and confirmed repeatedly by experimental means. In its simplest construction, this proposition was mathematically substantiated by J. Bernoulli in the theorem bearing his name. This theorem related to the case, or more precisely, to a group of cases, in which, when passing from experiment to experiment, the probability—the ratio of the number of chances favorable to the occurrence of the event to the total number of chances of the event under consideration—did not change (e.g., when balls of black or white color are extracted from an urn and placed back). In a more general form, closer to the conditions in which the phenomena of human life take place, namely, when the probabilities of events change from one experiment or case to another (e.g., when balls are extracted not from one, but from a multitude of urns with a different proportion of white and black balls), the correctness of the same proposition was proven in Poisson's theorem; he also gave this phenomenon the name (not entirely successful in the opinion of A. A. Chuprov)—'Law of Large Numbers.' In the experimental sense, the Law of Large Numbers has been subjected to repeated verification. Thus, Quetelet repeated 4,096 extractions from a vessel containing black and white balls in equal proportion; as a result, he obtained 2,066 extractions of white balls and 2,030 extractions of black balls; Westergaard, with 10,000 extractions, obtained 5,011 balls of one color and 4,989 balls of the other color; Buffon, with 4,040 tosses of one coin, obtained 2,048 'heads' and 1,992 'tails.' Jevons, with 1,024 simultaneous tosses of ten coins, repeated twice, obtained the following results: 'Heads' ... Total 'Tails' Theoretical output numbers... 1st series of experiments... 2nd series of experiments... Average of 1st and 2nd series. In a general, non-mathematical form, the Law of Large Numbers states: in a large number, constituting the result of a statistical mass observation, such regularities appear (regularities both in the structure of a known social mass and in the occurrence of actions and events) which cannot be discerned in arbitrarily fractional parts of the studied mass (G. Mayr).