336 PROBLEMS ON CAUSES. [CHAP. XX. 



and it may be added, as a final confirmation of their correctness, 

 that the above results become identical when m = pqr. 



9. The following problem is a generalization of Problem I., 

 and its solution, though necessarily more complex, is obtained by 

 a similar analysis. 



PROBLEM VI. If an event can only happen as a conse- 

 quence of one or more of certain causes A l9 A 29 . . A n9 and if 

 generally Ci represent the probability of the cause A i9 and pi the 

 probability that if the cause AI exist, the event E will occur, 

 then the series of values of c f and pt being given, required the 

 probability of the event E* 



Let the causes A l9 A Z9 . . A n be represented by x l9 x Z9 . . #; 

 and the event E by z. 



Then we have generally, 



Prob. Xi = ci, Prob. xiz = ci p^ 



Further, the condition that E can only happen in connexion with 

 some one or more of the causes A A z ,..A n establishes the logi- 

 cal condition, 



r(l-^)(l-^)..(l-^) = 0. (1) 



* It may be proper to .remark, that the above problem was proposed to the 

 notice of mathematicians by the author in the Cambridge and Dublin Mathema- 

 tical Journal, Nov. 1851, accompanied by the subjoined observations : 



" The motives which have led me, after much consideration, to adopt, with 

 reference to this question, a course unusual in the present day, and not upon 

 slight grounds to be revived, are the following : First, I propose the question 

 as a test of the sufficiency of received methods. Secondly, I anticipate that its 

 discussion will in some measure add to our knowledge of an important branch 

 of pure analysis. However, it is upon the former of these grounds alone that I 

 desire to rest my apology. 



" While hoping that some may be found who, without departing from the line 

 of their previous studies, may deem this question worthy of their attention, I 

 wholly disclaim the notion of its being offered as a trial of personal skill or 

 knowledge, but desire that it may be viewed solely with reference to those pub- 

 lic and scientific ends for the sake of which alone it is proposed." 



The author thinks it right to add, that the publication of the above problem 

 led to some interesting private correspondence, but did not elicit a solution. 



