360 PROBLEMS ON CAUSES. [CHAP. XX. 



X l X* . . X n t + X } (1 - t) = X l Xy . . X n t + X 2 (1 - t) 



p p 



x l x z . . x n t + I - t 



From the forms of the above equations it is evident that we 

 have #] = # 2 = # Replace then each of these quantities by #, 

 and the system becomes 



p a 



Prob. w 



g,^ 



x n t+ l-t 9 

 from which we readily deduce 



Prob. w = Prob. x 1 x z . . x n = a + (jp - a) I y- j 

 If in this result we change n into n + 1, we get 



Prob. Xi x 2 . . x n+} = a + (p - a) f jT") ' 

 Hence we find 



as the expression of the probability that if the phenomenon be n 

 times repeated, it will also present itself the n + I th time. By the 

 method of Chapter XIX. it is found that a cannot exceed p in 

 value. 



The following verifications are obvious : 



1st. If a =0, the expression reduces to p, as it ought to do. 

 For when it is certain that no permanent cause exists, the suc- 

 cessive occurrences of the phenomenon are independent. 



2nd. If p = 1, the expression becomes 1, as it ought to do. 



3rd. If p = , the expression becomes 1, unless a = 0. If the 

 probability of a phenomenon is equal to the probability that there 



