CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 263 



prior assumption as to the nature of the balls, and their relations, 

 or freedom from relations, of form, colour, structure, &c. It is 

 founded upon our total ignorance of all these things. Probabi- 

 lity always has reference to the state of our actual knowledge, 

 and its numerical value varies with varying information. 



PROPOSITION III. 



11, To determine in any question of probabilities the logical 

 connexion of the qucesitum with the data; that is, to assign the event 

 whose probability is sought, as a logical function of the event whose 

 probabilities are given. 



Let S 9 T, &c., represent any compound events whose pro- 

 babilities are given, S and T being in expression known func- 

 tions of the symbols x, y, z, &c., representing simple events. 

 Similarly let W represent any event whose probability is sought, 

 W being also a known function of #, y, z, &c. As S, T, . . W 

 must satisfy the fundamental law of duality, we are permitted 

 to replace them by single logical symbols, s, t, . . w. Assume 



then 



s = S, t=T, w = W. 



These, by the definition of S, T, . . W, will be a series of 

 logical equations connecting the symbols s, t, . . w, with the sym- 

 bols x, y, z . . 



By the methods of the Calculus of Logic we can eliminate 

 from the above system any of the symbols #, y, z, . . , repre- 

 senting events whose probabilities are not given, and determine 

 w as a developed function of s, t, &c., and of such of the symbols 

 x, y, z, &c., if any such there be, as correspond to events whose 

 probabilities are given. The result will be of the form 



where A, B, C, and D comprise among them all the possible 

 constituents which can be formed from the symbols s, , &c., i. e. 

 from all the symbols representing events whose probabilities are 

 given. 



The above will evidently be the complete expression of the 

 relation sought. For it fully determines the event W 9 repre- 



