316 OF STATISTICAL CONDITIONS. [CHAP. XIX. 



for all possible problems in which the data are the same, how- 

 ever the qu&sita of those problems may vary. The nature of 

 that event, or combination of events, whose probability is sought, 

 will affect only the form of the function in which the determined 

 values of s l s^.. s n are to be substituted. 

 We have from (9) 



Whence 

 Or, 



i - X = (i-pA) (1 -M) (i -pnX) ; (10) 



from which equation the value of X is to be determined. 



Supposing this value determined, the value of Prob. x l will be 



__ /?iX- (1-c)/?!/^. . p n \ n 



i - (i ~- pi\) (i 



or, on reduction by (10), 



Prob.#! =P! - (1 - c) ji/? 2 ..p ra X n - 1 . (11) 



Let us next seek the conditions which must be fulfilled 

 among the constants p l9 p 25 /?nj and the limits of the value of 

 Prob. #! 



As there is but one term with the coefficient - , there is but 

 one condition among the constants, viz., 



Minor limit, (1 - $j) (1 - s 2 ) . . (1 - s n ) < 0. 

 Or, n (1 - Sl ) +n(l- s 2 ) ..+n(l- s n ) - (w- 1) (!)< 0. 

 Or, w(l)-(*i)-w(*2) ..-()< 0. 



Whence p l + p 2 . . + p n > 1, 



the condition required. 



The major limit of Prob. x l is the major limit of the sum of 



those constituents whose coefficients are 1 or - . But that sum is s x . 



Hence, 



Major limit, Prob. x l = major limit Si = p^ . 



