386 INDUCTION. 



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 pro- 

 duce 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 ei- 

 ther 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 antecedent probabilities. 



Case II. Reversing 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 proba- 

 bilities that if they did exist they would produce the effect. 



Case III. The third case, that in which the causes are unlike in both re- 

 spects, 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 proportional 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. Therefore, the probability 

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

 the cause, multiplied by the probability that if it existed it would produce 

 M. Which was to be demonstrated. 



Or we may prove the third case as we proved the first and second. Let 

 A be twice as frequent as B, and let them also be unequally likely, when 

 they exist, to produce M ; let A produce it twice in fsiH^* times, B thrice in ^ 

 four times. The antecedent probability of A is to that of B as 2 to 1 ; 

 the probabilities of their producing M are as 2 to 3 ; the product of these 

 ratios is the ratio of 4 to 3 ; and this will be the ratio of the probabilities 

 that A or B was the producing cause in the given instance. For, since A 

 is twice as frequent as B, out of twelve cases in which one or other exists, 

 A exists in 8 and B in 4. But of its eight cases. A, by the supposition, 

 produces M in only 4, while B of its four cases produces M in 3. M, there- 

 fore, is only produced at all in seven of the twelve cases ; but in four of 

 these it is produced by A, in three by B ; hence the probabilities of its be- 

 ing produced by A and by B are as 4 to 3, and are expressed by the frac- 

 tions ^ and -f-. Which was to be demonstrated. 



§ 6. It remains to examine the bearing of the doctrine of chances on 



