274 GENERAL METHOD IN PROBABILITIES. [CHAP. XVII. 



The proposition which affirms that some one of these must occur 

 will be expressed by the equation 



#! + t 2 . . + t n = 1 ; 



and, as when we pass in accordance with the reasoning of the 

 last section to numerical probabilities, the same equation remains 

 true in form, and as the probabilities t l9 2 . . t n are equal, we 



have 



nt, = i, 



whence f x = -, and similarly t z = -, t n = -. Suppose it then re- 

 quired to determine the probability that some one event of the 

 partial series 2 > t* - t m will occur, we have for the expression 

 required 



t l + t% . . + t m = - + - . . to m terms 

 n n 



m 



Hence, therefore, if there are m cases favourable . to the occur- 

 rence of a particular alternation of events out of n possible and 

 equally probable cases, the probability of the occurrence of that 



?// 



alternation will be expressed by the fraction . 



Now the occurrence of any event which may happen in diffe- 

 rent equally possible ways is really equivalent to the occurrence 

 of an alternation, i. e., of some one out of a set of alternatives. 

 Hence the probability of the occurrence of any event may be 

 expressed by a fraction whose numerator represents the number 

 of cases favourable to its occurrence, and denominator the total 

 number of equally possible cases. But this is the rigorous nume- 

 rical definition of the measure of probability. That definition is 

 therefore involved in the more peculiarly logical definition, the 

 consequences of which we have endeavoured to trace. 



20. From the above investigations it clearly appears, 1st, 

 that whether we set out from the ordinary numerical definition 

 of the measure of probability, or from the definition which assigns 

 to the numerical measure of probability such a law of value as 

 shall establish a formal identity between the logical expressions 



