74 INDUCTION. 



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 pro 

 duce it twice in four times, B thrice in four times. The ante 

 cedent probability of A is to that of B as 2 to 1 ; the proba 

 bilities 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, 

 therefore, 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 being produced by A and by B are as 

 4 to 3, and are expressed by the fractions f and f-. Which was 

 to be demonstrated. 



6. It remains to examine the bearing of the doctrine of 

 chances on the peculiar problem which occupied us in the pre 

 ceding chapter, namely, how to distinguish coincidences which 

 are casual from those which are the result of law ; from those 

 in which the facts which accompany or follow one another are 

 somehow connected through causation. 



The doctrine of chances affords means by which, if we knew 

 the average number of coincidences to be looked for between 

 two phenomena connected only casually, we could determine 

 how often any given deviation from that average will occur by 

 chance. If the probability of any casual coincidence, 



considered in itself, be , the probability that the same 



w&amp;gt; 



coincidence will be repeated n times in succession is . For 



m 



example, in one throw of a die the probability of ace being 

 -; the probability of throwing ace twice in succession 



will be 1 divided by the square of 6, or . For ace is 



