280 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



In the expression of this development, the coefficient - has been 

 made to replace every equivalent form (X. 6). Here we have 

 V= xy(\ - s) + x(l - y)s + (1- x)ys + (1 - x) (1 - y) (1 -*); 

 whence, passing from Logic to Algebra, 

 xy (1 - s) + #(1 -y)s _xy (I -s) + (I - x 



p ~Y~ 



r 

 - x)ys +(!-)(!- y) (1 - s). 



p , _ _ xy (1 - s) 

 ~ 



from which we readily deduce 



whence we have 



-r , . 



for the value sought. 



If in the same way we seek the probability that if A and B 

 agree in their statement, that statement will be false, we must 

 replace the second equation of the system (1) by the following, 



viz.: 



(l-x)(\-y)~w; 



the final logical equation will then be 



w=*-xys + Qxy(l-s) + Ox (1 - y) s + -x (I- y) (l-s) 



+ 0(1- x)ys + 1 (l-x)y(l-s) + -(1 - x) (1 - y)s 



+ (1-*)(1-2,)(1-.); (4) 

 whence, proceeding as before, we finally deduce 



2 



Wherefore we have 



(5) 



