35^ 



INDUCTION. 



produced twice by A, once only by 

 B. Consequently, when the ante- 

 dent 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 pre- 

 ceded. 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 pro- 

 portional to the product of the two 

 quantities, the product being the only 

 function of the two which obeys that 

 law of variation. Therefore the pro- 

 bability that M was produced by 

 either cause is as the antecedent pro- 

 bability 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 pro- 

 duce it twice in four times, B thrice in 

 four times. The antecedent probabi- 

 lity of A is to that of B as 2 to i ; 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, 

 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 ^ 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 preceding 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 devia- 

 tion from that average will occur by 

 chance. If the probability of any 

 casual coincidence, considered in it- 

 self, be -, the probability that the 

 same coincidence will be repeated n 

 times in succession is -—. For ex- 

 ample, in one throw of a die the pro- 

 bability of ace being -^, the probabi- 

 lity of throwing ace twice in succession 

 will be I divided by the square of 6, or 



-. For ace is thrown at the first 

 36 



throw once in six, or six in thirty- 

 six times, and of those six, the die 

 being cast again, ace will be thrown 

 but once ; being altogether once in 

 thirty-six times. The chance of the 

 same cast three times successively is, 



by a similar reasoning, ^ or — - ; that 



is, the event will happen, on a large 

 average, only once in two hundred 

 and sixteen throws. 



We have thus a rule by whicn to 

 estimate the probability that any 

 given series of coincidences arises 

 from chance, provided we can mea- 

 sure correctly the probability of a 

 single coincidence. If we can obtain 

 an equally precise expression for the 

 probability that the same series of 

 coincidences arises from causation, 

 we should only have to compare the 

 numbers. This, however, can rarely 

 be done. Let us see what degree 



