CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 279 



4. Ex. 3. The probability that a witness A speaks the truth 

 is p, the probability that another witness B speaks the truth is q, 

 and the probability that they disagree in a statement is r. What 

 is the probability that if they agree, their statement is true ? 



Let x represent the hypothesis that A speaks truth ; y that 

 B speaks truth ; then the hypothesis that A and B disagree in 

 then: statement will be represented by x (1 - y) + y (1 - x) ; the 

 hypothesis that they agree in statement by xy + (1 - x) (1 - y), 

 and the hypothesis that they agree in the truth by xy. Hence 

 we have the following data : 



Prob. x - p, Prob. y = q, Prob. x(l ~ y) + y (I - #) = r, 

 from which we are to determine 



Prob. xy 



Prob. xy + (1 - x) (1 - #)' 



But as Prob. x (1 - y) + y (Y- x) = r, it is evident that Prob. 

 xy + (1 - x) (1 - y) will be 1 - r ; we have therefore to seek 



Prob. xy 

 l-r 



Now the compound events concerned being in expression, 

 x (1 - y) + y (1 - x) and xy, let us assume 



*. -- J , (1) 



Our data then are Prob. x = p, Prob. y = q, Prob. 5 = r, and we 

 are to find Prob. w. 



The system (1) gives, on reduction, 



+ xy(l- w) + w(l -xy) = 0; 

 whence 



2xy- 1 

 -xys + xy(l-s)+Ox(l - y}s + -x (1 - y) (1 - s) 



-- 



