354 PROBLEMS ON CAUSES. [CHAP. XX. 



Probability that if some one ^ c r p r 



alone of the causes A^A z ..A n 1 __ ILL - c r p r _ 

 present itself, the event E f c r p r c r (l-p r ) 



Will follow. J fJL - C r p r V-C r (l-pr) 



Let it be observed that this case is quite different from the 

 well-known one in which the mutually exclusive character of 

 the causes A i9 . . A n is one of the elements of the data, expressing 

 a condition under which the very observations by which the pro- 

 babilities of A 19 A^ &c. are supposed to have been determined, 

 were made. 



Consider, lastly, the case in which (x l9 . . #) = Xi x z . . x n . 

 Here 



= Cipi ..c n p n _ = c l p l ..c n p n 



~ ~ ft (fJL + V - \ )' 



AT = 

 ~ 



Hence the following result 



Probability that if all the 



causes A ly . . A n con- _ 



spire, the event E will pi . . p n v"' 1 + (l -p^. .(l-p n ) ju 7 



follow. 



This expression assumes, as it ought to do, the value 1 when any 

 one of the quantities p\ 9 . .p n i& equal to 1. 



17. PROBLEM VIII. Certain causes A i9 A^..A n being so 

 restricted that they cannot all fail, but still can only occur in cer- 

 tain definite combinations denoted by the equation 



(A 19 A 2 . . A n ) = 1, 



and there being given the separate probabilities c l5 . . c n of the 

 said causes, and the corresponding probabilities p l9 . . p n that an 

 event E will follow if those respective causes are realized, re- 

 quired the probability of the event E. 



This problem differs from the one last considered in several 

 particulars, but chiefly in this, that the restriction denoted by the 

 equation (^4i, . . A n ) = 1, forms one of the data, and is supposed 



