328 PROBLEMS ON CAUSES. [CHAP. XX 



if the event B take place, the same event E will accompany it 

 is q. Required the probability that if the event A take place the 

 event B will take place, or vice versa, the probability that if B 

 take place, A will take place. 



Let us represent the event A by x 9 the event B by y, and the 

 event E by z. Then the data are 



Prob. x = a 9 Prob. y - b. 

 Prob. xz = ap 9 Prob. yz = bq. 

 Whence it is required to find 



Prob. xy Prob. xy 

 Prob. re Prob.^* 



Let a^ = s 9 yz = , ar# = w. 



Eliminating z, we have, on reduction, 



sx + ty + sy~t + xfs + xyw + (1 - xy) 10 = 0, 

 _ sx + ty + syJ+ xfs + xy 





= xyst -4- 



+ - xys~t+ Qxys~+ - xyst+ - xyst 



+ n x y~*t + n x y&t + Qxy~st + - xy~st 

 + xyst + Qxy~s~t + Qxy~s~f + Qxy'sT. (1) 



Hence, passing from Logic to Algebra, 



xyst -f xys~i 



cc, y, s, and f being determined by the system of equations 

 xyst + xys7+ xysJ+ xy~s1 _ xyst + xy~st + xys~t+ xys7 



~~aT "T" 



xyst + xyst xyst + Icyst 

 ap bq 



= xyst + xysl + xyst + xys~t + xy'st + xy~s~t + xy~st = F. 



