254 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



tion, . Of the m cases favourable to the first event, let / 



m + n 



cases be favourable to the conjunction of the first and second 

 events, then, by definition, is the probability that if the first 



event happen, the second also will happen. Multiplying these 

 fractions together, we have 



m l_ I 



m + n m m + n 



But the resulting fraction has for its numerator the num- 



m + n 



ber of cases favourable to the conjunction of events, and for its 

 denominator, the number m + n of possible cases. Therefore, 

 it represents the probability of the joint occurrence of the two 

 events. 



Hence, if p be the probability of any event x, and q the pro- 

 bability that if x occur y will occur, the probability of the con- 

 junction xy will be pq. 



III. The probability that if an event x occur, the event y will 

 occur, is a fraction whose numerator is the probability of their 

 joint occurrence, and denominator the probability of the occur- 

 rence of the event x. 



This is an immediate consequence of Principle 2nd. 



IV. The probability of the occurrence of some one of a series 

 of exclusive events is equal to the sum of their separate proba- 

 bilities. 



For let n be the number of possible cases ; m^ the number of 

 those cases favourable to the first event ; m 2 the number of cases 

 favourable to the second, &c. Then the separate probabilities of 



the events are , , &c. Again, as the events are exclusive, 

 n n 



none of the cases favourable to one of them is favourable to 

 another; and, therefore, the number of cases favourable to some 

 one of the series will be m l + m z . . , and the probability of some 



one of the series happening will be - . But this is the 



sum of the previous fractions, - , , &c. Whence the prin- 

 ciple is manifest. 



