278 ELEMENTARY ILLUSTRATIONS. [CHAP. XVIII. 



3. Ex. 2. The probability that one or both of two events 

 happen is p, that one or both of them fail is q. What is the 

 probability that only one of these happens ? 



Let x and y represent the respective events, then the data 



are 



Prob.xy + x (1 -y) + (1 - x)y = p, 



Prob.a?(l -y) + (l-x)y + (1 - x) (1 -y) = y; 

 and we are to find 



Here all the events concerned being compound, assume 



Then eliminating x and y, and determining w as a developed 

 function of s and t, we find 



w = st + s (1 - *) + (1 - s) t + i (1 - 5) (1 - *). 



Hence ^ = s, C=0, V=st + s(l-t) + (1 -s)t = s + (1 -s)t, 

 V-s^ Vf=t- 9 and the equations of the General Rule (XVII. 17) 

 become 



Prob. 



~' H , . 0) 



w 



(l -s)t' 

 whence we find, on eliminating s and t, 



Prob. ic =p + g - 1. 



Hence p + q - 1 is the measure of the probability sought. This 

 result may be verified as follows : Since p is the probability that 

 one or both of the given events occur, 1 - p will be the proba- 

 bility that they both fail ; and since q is the probability that one 

 or both fail, 1 - q is the probability that they both happen. 

 Hence 1 - p + 1 - #, or 2 -p - q, is the probability that they 

 either both happen or both fail. But the only remaining alter- 

 native which is possible is that one alone of the events happens. 

 Hence the probability of this occurrence is 1 - (2 - p - q) 9 or 

 p -f q - 1 , as above. 



