The Law of Large Numbers

at the same time, the mean chance of the contrary event F will be the sum of the fractions 1− p1, −− p2, . . . 1− pμ , divided by μ; and by designating it by q′, one will have p′+q′ = 1. This being, the one of the general propositions, which we wish to consider, consists in this that if one calls m and n the numbers of times that E and F will arrive or are arrived during the series of these trials, the ratios of m and n to the total number μ or m+n, will be, very nearly and with a very great probability, the values of the mean chances p′ and q′, and reciprocally, p′ and q′ will be the approximate values of μ and n μ . When these ratios will have been deduced from a long series of trials, they will make known therefore the mean chances p′ and q′, likewise they determine, by the rule of no 49, the same chances p and q of E and F, when they are constants. But in order that these approximate values of p′ and q′ are able to serve, also by approximation, to evaluate the numbers of times that E and F will arrive in a new series of a great number of trials, it is necessary that it be certain, or at least very probable, that the mean chances of E and F will be exactly, or quite nearly the same, for this second series, and for the first. Now, it is this which holds effectively by virtue of another general proposition of which here is the enunciation. I suppose that by the nature of the events E and F, the one which will arrive at each trial is able to be due to one of the causes C1, C2, C3, . . .Cν , of which ν is the number, which is mutually exclusive, and which I will regard first as equally possible.