260 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



tion whose numerator is the probability of the joint occurrence 

 of V and F', and denominator the probability of the occurrence 

 of V. 



Now the expression of that event, or state of things, which is 

 constituted by the joint occurrence of the events Fand F', will 

 be formed by multiplying together the expressions F and V ac- 

 cording to the rules of the Calculus of Logic ; since whatever 

 constituents are found in both V and V will appear in the pro- 

 duct, and no others. Again, by what has just been shown, the 

 probability of the event represented by that product will be de- 

 termined by changing therein #, y, z into /?, q, r, . . Hence the 

 numerator sought will be what \_W~\ by definition represents. 

 And the denominator will be [F], wherefore 



[FF1 

 Probability that if F occur, V will occur with it = -*. 



9. For example, if the probabilities of the simple events 

 x, y, z are p, q, r respectively, and it is required to find the pro- 

 bability that if either x or y occur, then either y or z will occur, 

 we have for the logical expressions of the antecedent and conse- 

 quent 



1st. Either x or y occurs, x (1 - y) + y (1 - x). 

 2nd. Either y or z occurs, y (1 - z) + z (1 - y). 



If now we multiply these two expressions together according to 

 the rules of the Calculus of Logic, we shall have for the expres- 

 sion of the concurrence of antecedent and consequent, 



xz(\-y)+ y (\-x)(\-z). 



Changing in this result #, y, z into p, q, r, and similarly trans- 

 forming the expression of the antecedent, we find for the proba- 

 bility sought the value 



The special function of the calculus, in a case like the above, is 

 to supply the office of the reason in determining what are the 

 conjunctures involved at once in the consequent and the ante- 

 cedent. But the advantage of this application is almost entirely 



