HAP. XVIII.] ELEMENTARY ILLUSTRATIONS. 277 



y = - = ux + - u (1 - x) + (1 - u) x + - (1 - u) (1 - x). 



Hence (XVII. 17) we find 



V = ux + (1 - u) x + (1 - u) (1 - z), 

 V x = ux + (1 - u) x = x, V u = ux\ 



and the equations of the General Rule, viz., 



Prob. y = 



A + cC 



V 



become, on substitution, and observing that A = ux, C= (1 - u) 

 (1 - #), and that F reduces to # + (1 - u) (1 - #), 



f V7 1 



- = - = *H-0-)(l-*). (3) 



UX + C(l- U) (1 - X) ... 



* n i-)0-x) ' (4) 



from which we readily deduce, by elimination of x and u 9 



= q + c(l-p). (5) 



In this result c represents the unknown probability that if the 

 event (1 -u) (1 - x) happen, the event y will happen. Now 

 (I -u) (!-#) = (!- xy) (1 - #) = 1 - x, on actual multiplication. 

 Hence c is the unknown probability that if it do not thunder, it 

 will hail. 



The general solution (5) may therefore be interpreted as fol- 

 lows : The probability that it hails is equal to the probability 

 that it thunders and hails, q, together with the probability that it 

 does not thunder, 1 -jo, multiplied by the probability c, that if it 

 does not thunder it will hail. And common reasoning verifies 

 this result. 



If c cannot be numerically determined, we find, on assigning 

 to it the limiting values and 1, the following limits of Prob. y, 

 viz. : 



Inferior limit = q. 

 Superior limit = q + I - p. 



