82 INDUCTION. 



those in which the facts which accompany or follow 

 one another are somehow connected through causa- 

 tion. 



5. 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 coincidence 



will be repeated n times in succession is . For ex- 

 ample, in one throw of a die the probability of ace 

 being g ; the probability of throwing ace twice in 

 succession will be 1 divided by the square of 6, or 

 oTj. For ace is thrown at the first 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 grg : that is, the event will happen, 



on a large average, only once in two hundred and 

 sixteen throws. 



We have thus a rule by which to estimate the pro- 

 bability that any given series of coincidences arises 

 from chance ; provided we can measure correctly the 

 probability of a single coincidence. If we could 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, 



