282 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



The symbol x to the visitation of fever. 

 y ,, cholera. 



z defective sanitary condition. 



The events whose probabilities are given are then denoted by 

 x, y, and (I - x) (1 - y) (1 - ,?), the event whose probability is 

 sought is z. Assume then, 



then our data are, 



Prob. x = p 9 Prob. y = q, Prob. w = r, 

 and we are to find Prob. z. Now (1) gives 



+ ^(l-x)yw + -(l 



+ 0-a)(i-y)(i-0- 0) 



The value of Fdeduced from the above is 



F= xy (1 - w) + x (1 - y) (1 - w) + (1 - x) y (1 - w) 

 + (l-x)(l-y)w + (l-x)(l-y)(l-w)=l 

 and similarly reducing V X9 V y , V w , we get 



V x = x(l-w), V s = y(l-w) 9 V w ~ 

 furnishing the algebraic equations 



.-)(l-y). (2) 



As respects those terms of the development characterized by 

 the coefficients -, I shall, instead of collecting them into a single 



term, present them, for the sake of variety (XVII. 18), in the 

 form 



w )3 (3) 



the value of Prob. z will then be 



