CHAP. XX.] PROBLEMS ON CAUSES. 353 



apprehend that the same method is generally applicable and suf- 

 ficient. But this is a question upon which a further degree of 

 light is desirable. 



To verify the above results, suppose (x . .#) = 1, which is 

 virtually the case considered in the previous problem. Now the 

 development of 1 gives all possible constituents of the symbols 

 x l9 . . x n . Proceeding then according to the Bule, we find 



M, = 7 - t , - 1 = _ - 1 by (15). 



- ft jh) . (ft- C npn) fJL + V - 1 



Ni = + l---- -1. 



{v-C^l-p,)} .. {v-C n (l -/?)} fjL + V~l 



Substituting in (14) we find 



Prob.z= 1 -v, 



which agrees with the previous solution. 



Again, let $ (#1, . . x n ) = x i9 which, after development and sup- 

 pression of the factors x t9 . . x n , gives x 1 (x z + 1) . . (x n + 1), whence 

 we find 



^ -= C ^' .by (15). 



) ..(fi-c n p n ) ft + v- 1 



{v-c^l-p} .. {v-c n (l -/?)} n+v-l 

 Substituting, we have 



Probability that if the event AI occur, E will occur = p lt 



And this result is verified by the data. Similar verifications 

 might easily be added. 



Let us examine the case in which 



Here we find 



fJL ~ C^ fjl - C 



_ , 



' 



_ 

 V - C, (1 - /?x) ' V - C n (1 -p n ) ' 



whence we have the following result 



2 A 



