CHAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 283 



Prob = o-^o-yx^^+^o- 



l-w + w(l-x) 

 From (2) and (4) we deduce 



(1 - p - r) (1 - q - r) 

 Prob. z = ^ F * ++ 



as the expression of the probability required. If in this result 

 we make c = 0, and d = 0, we find for an inferior limit of its value 



- p-i - -; and if we make c = 1, c'= 1, we obtain 



for its superior limit 1 - r. 



6. It appears from inspection of this solution, that the pre- 

 mises chosen were exceedingly defective. The constants c and 

 d indicate this, and the corresponding terms (3) of the final 

 logical equation show how the deficiency is to be supplied. 

 Thus, since 



we learn that c is the probability that if any house was visited by 

 fever its sanitary condition is defective, and that c is the proba- 

 bility that if any house was visited by cholera without fever, its 

 sanitary condition was defective. 



If the terms of the logical development affected by the coeffi- 



cient - had been collected together as in the direct statement of 



the general rule, the final solution would have assumed the fol- 

 lowing form : 



-n i- 



Prob. z 





p + q - -2 



c here representing the probability that if a house was visited by 

 either or both of the diseases mentioned, its sanitary condition 

 was defective. This result is perfectly consistent with the former 

 one, and indeed the necessary equivalence of the different forms 

 of solution presented in such cases may be formally established. 



The above solution may be verified in particular cases. Thus, 

 taking the second form, if c= I we find Prob. z = 1 - r, a correct 

 result. For if the presence of either fever or cholera certainly 



