270 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



Let A + B + C = F, and let V 8 represent the aggregate of 

 those constituents in V which contain s as a factor, V t of those 

 which contain t as a factor, and thus for all the symbols whose 

 probabilities are given. 



Then, passing from Logic to Algebra, form the equations 



|-f^ 



-o A + cC 



Prob. 10 - F , (2) 



from (1) determine s, t, &c. as functions of/?, ^, &c., and sub- 

 stitute their values in (2). The result will express the solution 

 required. 



Or form the symmetrical system of equations 



V._V, A + cC V 



J~~q' ~iT T 



where u represents the probability sought. 



If c appear in the solution, its interpretation will be 



_ Prob. Cw 

 Prob. c ' 



and this interpretation indicates the nature of the experience 

 which is necessary for its discovery. 



CASE II. When some of the events are conditioned. 

 If there be given the probability p that if the event X occur, 

 the event Y will occur, and if the probability of the antecedent 

 X be not given, resolve the proposition into the two following, 

 viz.: 



Probability of X = c, 

 Probability of Xy = cp. 



If the quaesitum be the probability that if the event W occur, 

 the event Z will occur, determine separately, by the previous 

 case, the terms of the fraction 



Prob. WZ 

 Prob. W ' 



and the fraction itself will express the probability sought. 



