258 PROCEEDINGS OF THE AMERICAN ACADEMY 



Now p',q' = p' — p' ,q' ■==. q' — q' ,p' 



p' ,q' = q' — p' ,q' == jj' — q' , p' 



And p',q> ==p' — p\, q' =p q' — q>-, p> 



p',q' ==q' — q' pl p' =zp' — p'- t q> 



Then let 



A-r-Px-r- yjZ 



d -, z,y,2-}-a,y,z-fa:,y,« + a,y,; 



Jj ^ ( ^'^-- 1- X ,y 





i — y, z 



7) _ _ / — x iV -> z 

 D -r-4r»-r x , y 



And we have 



= | p+F (I_ ? )_ a+F) (|_,_*(I^>)) 



= ?+ r(^)-(i+r)(-/>-c(i- 9 )) 

 = ? +r(^)-(i + r)( ? -i| P ) 



jBic. 2. (See Boole, p. 276.) Given r and </ ; to find p. 



p = r ; q =p r -\- v , (1 — q) because p is interpretable. 



Ans. The required proportion lies somewhere between the pro- 

 portion of days upon which it both hails and thunders, and that added 

 to one minus the proportion of days when it thunders. 



Ex. 3. (See Boole, p. 279.) Given, out of the number of questions 

 put to two witnesses, and answered by yes or no, the proportion that 

 each answers truly, and the proportion of those their answers to which 

 disagree. Required, out of those wherein they agree, the proportion 

 they answer truly and the proportion they answer falsely. 



