CALCULATION OF CHANCES. 77 



an individual fact. The signs or evidences by which 

 a fact is usually proved, are some of its consequences: 

 and the inquiry hinges upon determining what cause 

 is most likely to have produced a given effect. The 

 theorem applicable to such investigations is the Sixth 

 Principle in Laplace's Essai Philosophique sur les 

 Probabilite's, which is described by him as cc the fun- 

 damental principle of that branch of the Analysis of 

 Chances, which consists in ascending from events to 

 their causes*," 



Given an effect to be accounted for, and there 

 being several causes which might have produced it, 

 but of the presence of which, in the particular case, 

 nothing is known; the probability that the effect was 

 produced by any one of these causes is as the ante- 

 cedent probability of the cause, multiplied by the proba- 

 bility that the cause, if it existed, would have produced 

 the given effect. 



Let M be the effect, and A, B, two causes, by either 

 which it might have been produced. To find the pro- 

 bability that it was produced by the one and not by 

 the other, ascertain which of the two is most likely 

 to have existed, and which of them, if it did exist, 

 was most likely to produce the effect M: the pro- 

 bability sought is a compound of these two proba- 

 bilities. 



CASE I. Let the causes be both alike in the second 

 respect ; either A or B, when it exists, being sup- 

 posed equally likely (or equally certain) to produce M ; 

 but let A be in itself twice as likely as B to exist, 

 that is, twice as frequent a phenomenon. Then it is 



* Pp. 18, 19. The theorem is not stated by Laplace in the 

 exact terms in which I have stated it; but the identity of import 

 of the two modes of expression is easily demonstrable. 



