CHAP. XX.] PROBLEMS ON CAUSES. 345 



_ 0* - fr) . . Q - ) 



M- ^-1 



wherein 



(1 - ti) . 



P^-**Jr^fe-^ 



Hence, 



l- W =(l- Cl )..(l-O, 



.-. tc = 1 - (1 - c x ) . . (1 - O, 

 equivalent to the a priori determination above. 

 2nd. Let p l = 0, p 2 = 0, p n = 0, then (19) gives 



JU - tt = jU, 

 .'. M= 0, 



as it evidently ought to be. 



3rd. Let c : , c 2 ..c n be small quantities, so that their squares 

 and products may be neglected. Then developing the second 

 members of the equation (19),* 



fJL n - (C.p, + C 2 p z . . + C n p n ) fl n - 1 



>-.- -r- 



.'. U = G!/?! + C 2 p z . . + C n p n . 



Now this is what the solution would be were the causes 

 AI, A z . . A n mutually exclusive. But the smaller the proba- 

 bilities of those causes, the more do they approach the condition 

 of being mutually exclusive, since the smaller is the probability of 

 any concurrence among them. Hence the result above obtained 

 will undoubtedly be the limiting form of the expression for the 

 probability of E. 



4th. In the particular case of n = 2, we may readily elimi- 

 nate /i from the general solution. The result is 



(U - CijPi) (U - C 2 j9 2 ) _ { 1 - d (1 - ffQ - U\ { 1 - C z (1 - j? 2 ) - U) 



- u I -u 



which agrees with the particular solution before obtained for this 

 case, Problem i. 



Though by the system (19), the solution is in general made 

 to depend upon the solution of an equation of a high order, its 



