OF THE CALCULATION OF CHANCES. 73 



frequent a phenomenon. Then it is twice as likely to have 

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

 duced 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 : therefore in 300 times that M is 

 produced, A was the producing cause 200 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 probability 

 as to which actually produced it, is in the ratio of their ante- 

 cedent probabilities. 



CASE II. Eeversing the last hypothesis, let us suppose 

 that the causes are equally frequent, equally likely to have 

 existed, but not equally likely, if they did exist, to produce M : 

 that in three times in which A occurs, it produces that effect 

 twice, while B, in three times, produces it only once. 

 Since the two causes are equally frequent in their occurrence ; 

 in every six times that either one or the other exists, A exists 

 three times and B three times. A, of its three times, produces 

 M in two ; B, of its three times, produces M in one. Thus, 

 in the whole six times, M is only produced thrice; but of that 

 thrice it is produced twice by A, once only by B. Con- 

 sequently, when the antecedent probabilities of the causes are 

 equal, the chances that the effect was produced by them are in 

 the ratio of the probabilities that if they did exist they would 

 produce the effect. 



CASE III. The third case, that in which the causes are 

 unlike in both respects, is solved by what has preceded. For, 

 when a quantity depends on two other quantities, in such a 

 manner that while either of them remains constant it is pro- 

 portional to the other, it must necessarily be proportional to 

 the product of the two quantities, the product being the only 

 function of the two which obeys that law of variation. There- 

 fore, the probability that M was produced by either cause, is as 

 the antecedent probability of the cause, multiplied by the pro- 

 bability that if it existed it would produce M. Which was to 

 be demonstrated. 



