344 PROBLEMS ON CAUSES. [CHAP. XX. 



increases, since ju. increases, and, with it, every factor contained. 

 Again, the negative term JJL - u diminishes with the increase of 

 u, as appears from its value deduced from (15), viz., 



(>! - u) . . (b n - u) 



(i -)-' 



Hence then, between the limits u = a l9 u = b i9 the first member 

 of ( 1 7) continuously increases, changing in so doing from a nega- 

 tive to a positive value. Wherefore, between the limits assigned, 

 there exists one value of u, and only one, by which the said 

 equation is satisfied. 



12. Collecting these results together, we arrive at the follow- 

 ing solution of the general problem. 



The probability of the event E will be that value of u de- 

 duced from the equation 



wherein 



. 



which (value) lies between the two sets of quantities, 



Cip\> c 2 p 29 . . c n p n and 1 - c t (1 -p\) 9 1 - c z (1 -pj . . 1 - c n (!-/>), 



the former set being its inferior, the latter its superior, limits. 



And it may further be inferred in the general case, as it has 

 been proved in the particular case of n = 2, that the value of u, 

 determined as above, will not exceed the quantity 



Cipi + c 2 p 2 . . + c n p n . 







13. Particular verifications are subjoined. 



1st. Let pi = 1, p 2 = 1, . . p n = 1. This is to suppose it cer- 

 tain, that if any one of the events A i9 A 2 . . A n9 happen, the 

 event E will happen. In this case, then, the probability of the 

 occurrence of E will simply be the probability that the events or 

 causes AH A 2 . A n do not all fail of occurring, and its expression 

 will therefore be 1 - (1 - c,) (1 - c 2 ) . . (1 - c n ). 



Now the general solution (19) gives 



