CHAP. XX.] PROBLEMS ON CAUSES. 363 



which, on replacing n by its value y , will be found to agree 



with (3)." 



Similar verifications might, it is probable, also be found for 

 the following results, obtained by the direct application of the 

 general method. 



The probability, under the same circumstances, that if, out of 

 n occasions, the event happen r times, and fail n - r times, it will 

 happen on the n + I th time is 



a + m (p - a] 



- la\ r - 1 





. . n(n-l) ..n-r+I ,, r 



wherein m = r-^ and I - -. 



1 .2..r n 



The probability of a permanent cause (r being less than n) 

 is 0. This is easily verified. 



If p be the probability of an event, and c the probability that 

 if it occur it will be due to a permanent cause ; the probability 

 after n successive observed occurrences that it will recur on the 

 n + I th similar occasion is 



c + (1 - c)x n 



wherein x 



20. It is remarkable that the solutions of the previous pro- 

 blems are void of any arbitrary element. "We should scarcely, 

 from the appearance of the data, have anticipated such a circum- 

 stance. It is, however, to be observed, that in all those problems 

 the probabilities of the causes involved are supposed to be known 

 a priori. In the absence of this assumed element of knowledge, 

 it seems probable that arbitrary constants would necessarily ap- 

 pear in the final solution. Some confirmation of this remark is 

 afforded by a class of problems to which considerable attention 

 has been directed, and which, in conclusion, I shall briefly 

 consider. 



