322 PROBLEMS ON CAUSES. [CHAP. XX. 



Let us represent 



The cause A by x. 

 The cause A 2 by y . 

 The effect E by z. 



Then we have the following numerical data : 



Prob. x = c 19 Prob. # = c 2 , 



Prob. xz = Ci_p M Prob. yz = C 2 p 2 . 



Again, it is provided that if the causes A 19 A 2 are both ab- 

 sent, the effect E does not occur ; whence we have the logical 

 equation 



(l-x)(l-y) = v(l-z). 

 Or, eliminating u, 



z(l-x)(l-y)~ 0. (2) 



Now assume, 



xz = s, yz = t. (3) 



Then, reducing these equations (VIII. 7), and connecting the 

 result with (2), 



xz(l-s)+s(l-xz) + yz(l-t)+t(l-yz) + z(l-x)(l-y)=Q. (4) 



From this equation, z must be determined as a developed 

 logical function of #, y, s, and , and its probability thence de- 

 duced by means of the data, 



Prob. x = GI , Prob. y = c^ Prob. s = c t pi , Prob. t = c z p 2 . (5) 



Now developing(4) with respect to z, and putting x for 1 - #, 

 ^ for 1 = y, and so on, we have 



(x s + so; + yi + ty + xy) z + (s + 2) z = 0, 



s + t-xs-sx-yt - ty - xy 



I , 1 _ 1 

 000 



1 - 1 -_ 1 - 



-f - stxy + stxy + -stxy-}- 

 00 



1- 1 _ 1_ 



M stxy -\ stxy + stxy -f s t xy 

 00 



+ Q7txy + Qstxy + O'stxy-}- Vstxy. (6) 



