CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 253 



CHAPTER XVII. 



DEMONSTRATION OF A GENERAL METHOD FOR THE SOLUTION OF 

 PROBLEMS IN THE THEORY OF PROBABILITIES. 



1. TT has been defined (XVI. 2), that "the measure of the 

 J- probability of an event is the ratio of the number of cases 

 favourable to that event, to the total number of cases favourable 

 or unfavourable, and all equally possible." In the following in- 

 vestigations the term probability will be used in the above sense 

 of " measure of probability." 



From the above definition we may deduce the following con- 

 clusions. 



I. When it is certain that an event will occur, the probability 

 of that event, in the above mathematical sense, is 1 . For the 

 cases which are favourable to the event, and the cases which are 

 possible, are in this instance the same. 



Hence, also, ifp be the probability that an event x will happen, 

 1 - p will be the probability that the said event will not happen. 

 To deduce this result directly from the definition, let m be the 

 number of cases favourable to the events, n the number of cases 

 possible, then n - m is the number of cases unfavourable to the 

 event x. Hence, by definition, 



- = probability that x will happen. 



= probability that x will not happen. 



But n - m m 



=1 =!-. 



n n 



II. The probability of the concurrence of any two events is 

 the product of the probability of either of those events by the 

 probability that if that event occur, the other will occur also. 



Let m be the number of cases favourable to the happening 

 of the first event, and n the number of equally possible cases un- 

 favourable to it; then the probability of the first event is, bydefini- 



