362 PROBLEMS ON CAUSES. [CHAP. XXi 



a + (- 



It is obvious that the value of this expression increases with the 

 value of n. 



I am indebted to a learned correspondent,* whose original 

 contributions to the theory of probabilities have already been re- 

 ferred to, for the following verification of the first of the above 

 results (3). 



" The whole a priori probability of the event (under the cir- 

 cumstances) being p, and the probability of some cause C which 

 would necessarily produce it, a, let x be the probability that it 

 will happen if no such cause as C exist. Then we have the 



equation 



p = a + (1 - a) x, 



whence _ p-a 



~T^~a 



Now the phenomenon observed is the occurrence of the event n 

 times. The a priori probability of this would be 



1 supposing C to exist, 



& supposing C not to exist ; 



whence the d posteriori probability that C exists is 



a + - 

 that C does not exist is 



(!-)* 



a + (1 - a) x n ' 

 Consequently the probability of another occurrence is 



a (1 - a) x n 



a + (1 - a) x* X + a + (1 - a) af X "' 



or a + (1 - a) # n+1 



a + (1 - a) x" ' 



* Professor Donkin. 



