358 PROBLEMS ON CAUSES. [CHAP. XX. 



processes, I do not presume to inquire. Its chief value undoubt- 

 edly consists in its power, in the mastery which it gives us over 

 questions which would apparently baffle the unassisted strength 

 of human reason. For this cause it has not been deemed super- 

 fluous to exhibit in this chapter its application to problems, some 

 of which may possibly be regarded as repulsive, from their diffi- 

 culty. without being recommended by any prospect of immediate 

 utility. Of the ulterior value of such speculations it is, I con- 

 ceive, impossible for us, at present, to form any decided judg- 

 ment. 



19. The following problem is of a much easier description 

 than the previous ones. 



PROBLEM X. The probability of the occurrence of a certain 

 natural phenomenon under given circumstances is p. Observation 

 has also recorded a probability a of the existence of a permanent 

 cause of that phenomenon, i.e. of a cause which would always pro- 

 duce the event under the circumstances supposed. What is the 

 probability that if the phenomenon is observed to occur n times in 

 succession under the given circumstances, it will occur the n + I th 

 time ? What also is the probability, after such observation, of the 

 existence of the permanent cause referred to ? 



FIRST CASE. Let t represent the existence of a permanent 

 cause, and x 19 x z . .#n+i the successive occurrences of the natural 

 phenomenon. 



If the permanent cause exist, the events x^ , x 2 . . x n+l are ne- 

 cessary consequences. Hence 



t = vx 19 t = vx 2 , &c., 

 and eliminating the indefinite symbols, 



Now we are to seek the probability that if the combination 

 #1 x*+ x n happen, the event # n+1 ';will happen, i. e. we are to seek 

 the value of the fraction 



Prob. X x . . x 



Prob. 

 We will first seek the value of Prob. Xi x z . . x n . 



