. Various related theorems about possibilities for events, among them Bernoulli’s theorem and Poisson’s theorem. For a rough illustration of the general idea, suppose that a tossed coin is equally likely to fall heads or tails. Then, the law says, the longer the series of tosses, the greater the probability that the frequency of heads will be within some given distance of 50 per cent (e.g. between 49 per cent and 51 per cent; it cannot be exactly 50 per cent except after an even number of tosses). The law is not itself a prediction. If we assume, on whatever grounds, that the coin will behave in certain ways, e.g. that it will not show any bias, then the law spells out what it is that we are assuming.
To see what lies behind this illustration, consider all possible results, in terms of heads and tails, that a series of tosses of a fair coin could yield. Then the longer the series the greater the proportion, among the possible results for that series, of results containing between 49 per cent and 51 per cent heads. See also Bayes’s Theorem.
I.Hacking, The Taming of Chance, Cambridge UP, 1990. (See chapter 12 for some historical material on Bernoulli, Poisson and Chebyshev, who developed the law.)
J.R.Lucas, The Concept of Probability, Clarendon, 1970, chapter 5. (Offers proof of Bernoulli’s theorem. See also W.C.Kneale, Probability and Induction, Oxford UP, 1949, § 29, who is clearer on what the theorem actually says.)
J.O.Wisdom, Foundations of Inference in Natural Science, Methuen, 1952, chapter 20. (Brief statement of some relevant theorems, with proof of one.)
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