CALCULATION" OF CHANCES. 323 



some of its consequences: and the inquiry hinges upon determining 

 what cause is mOv-^t likely to have produced a given effect. The theo- 

 rem applicable to such investigations is the Sixth Principle in Laplace's 

 Essai Philo;sophiquc sur les Frobahilif^s, which is described by him as 

 " tlie fundamental principle of that branch of the Analysis- of Chances, 

 which consists in ascending from e^•vnts 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 antecedent jnohahiUty 

 of the cause, >n/(/tij>/ied by the probability that the cause, if it existed, 

 would have produced the given effect. 



Let M be the effect, and A, 13, two causes, by either which it might 

 have been produced. To find the probability 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 probability sought is a compound 

 of these two probabilities. ■ , 



Case I. Let the causes be both alike in the second respect ; either 

 A or B, when it exists, being supposed 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 twice as 

 likely to have existed in this case, and to have been the cause which 

 produced M. 



For, since A exists in nature twice as often as B ; in any 300 cases 

 in which one or other existed, A has existed 200 times and B 100. 

 But either A or B must have existed wherever M is produced : there- 

 fore in 300 times that M is produced, A was the producing cause 20ft' 

 times, B only 100, that is, in the ratio of 2 to 1. Thus, then, if the 

 causes are alike in their capacity of producing the effect, the proba- 

 bility as to which actually produced it, is in the ratio of their arrtecedent 

 probabilities. " ^ "^ 



Case II. Reversing the last hypothesis, let us suppose that the 

 causes are equally frequent, equally likely to have existed, but not 

 equally hkely, if they did exist, to produce M : that in throe times that 

 A occurs, it produces that effect twice, while B, in three times, pro- 

 duces it only once. Since the two causes are equally frequent in their 

 occuiTence ; in every six times that either one or the otlrer eJcists, A 

 exists tlu-ee times and B three times. A, of its three times, produces 

 M in two ; B, of it* three times, produces M in one. Thus, in the 

 whole six times, M is only produced thiice ; but of that thrice it is 

 produced twice by- A, once only by B. Consequently, when the an- 

 tecedent probabilities of the causes are equal, the chances that the 

 effect was produco<l by them are in the ratio of the probabilities that 

 if they did exist they would produce the effect. 



Case III. The third cose, that in which the causes are unlike ia 

 Doth respects, is solved by what ha.s preceded. For, when a quantity 

 depend;* upon two other quantities, in Such a manner that while either 

 of them remains constant it is proportional to the other, it must neces- 

 sarily be proportional to the jiruduct of the two quantities, tl»e product 



♦ Pp. 18, 19. The theorem is not slated by Laplaco in iHe exact terms in which I have 

 stated it ; but the identity of import of the two modes of expression is easily demonstrable. 



