CHAP. XX.] PROBLEMS ON CAUSES. 355 



to be furnished by or to be accordant with the very experience 

 from which the knowledge of the numerical elements of the 

 problem is derived. 



Representing the events A 19 . . A n by x l9 . . x n respectively, 

 and the event E by 0, we have 



Prob. x r = c n Prob. x r z = c r p r . (1) 



Let us assume, generally, 



X T z = t n 



then combining the system of equations thus indicated with the 

 equations 



x\ - - ~%n = 0, (#1, ..#) = !, or = 1, 



furnished in the data, we ultimately find, as the developed ex- 

 pression of z 9 



z=2 (XT) + Oft J, . . l n 2(X), (2) 



where X represents in succession each constituent found in 0, 

 and T a similar series of constituents of the symbols fi, . . ; 

 S(X T) including only symmetrical constituents with reference 

 to the two sets of symbols. 



The method of reduction to be employed in the present case 

 is so similar to the one already exemplified in former problems, 

 that I shall merely exhibit the results to which it leads. We 

 find 



M+N. (4) 



with the relations 



M! M n Ni N n 



C\p\ C n p n Ci ( 1 - pi) C n ( 1 - 



Wherein M is formed by suppressing in (x ly . . x n ) all the fac- 

 tors #1, . . JP, and changing in the result x\ into m\ 9 x n into m n9 

 while N is formed by substituting in M. , HI for mi , &c. ; more- 

 over MI consists of that portion of M of which mi is a factor, 

 NI of that portion of N of which n\ is a factor ; and so on. 



Let us take, in illustration, the particular case in which the 

 causes A l . . A n are mutually exclusive. Here we have 



(X 19 . . X n ) = X l X 2 . . X n . . . + X n ^ . . ~X n _ v 



2 A 2 



