CHAP. XX.] PROBLEMS ON CAUSES. 371 



symbol 1 - Xi will represent the circumstance of the i th ball being 

 black. It is evident that the several constituents formed of the 

 entire set of symbols Xi , x z , . . # M will represent in like manner 

 the several possible constitutions of the system of balls with 

 respect to blackness and whiteness, and the number of such con- 

 stitutions being 2^, the probability of each will, in accordance 



with the hypothesis, be . This is the value which we should 

 find if we substituted in the expression of any constituent for 

 each of the symbols x i9 x 29 . . x^ the value -. Hence, then, the 



probability of any event which can be expressed as a series of 

 constituents of the above description, will be found by substi- 



. tuting in such expression the value - for each of the above 



2 



symbols. 



Now the larger p is, the less probable it is that any ball 

 which has been drawn and replaced will be drawn again. As JJL 

 approaches to infinity, this probability approaches to 0. And 

 this being the case, the state of the balls actually drawn can be 

 expressed as a logical function of m of the symbols x i9 .. # 2 . . # M , 

 and therefore, by development, as a series of constituents of the 

 said m symbols. Hence, therefore, its probability will be fonnd 

 by substituting for each of the symbols, whether in the unde- 

 veloped or the developed form, the value -. But this is the very 



substitution which it would be necessary, and which it would 

 suffice, to make, if the probability of a white ball at each drawing 



were known, a priori, to be - . 



It follows, therefore, that if the number of balls be infinite, 

 and all constitutions of the system equally probable, the proba- 

 bility of drawing m white balls in succession will be , and the 



probability of drawing m + 1 white balls in succession -^-; 

 whence the probability that after m white balls have been drawn, 

 the next drawing will furnish a white one, will be -. In other 



2s 2 



