CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 273 



For, in the first place, it is evident that, under this hypothesis, 

 the probability of the occurrence of some one of a set of mutually 

 exclusive events will be equal to the sum of the separate proba- 

 bilities of those events. Thus if the alternation in question con- 

 sist of n mutually exclusive events whose expressions are 



tf>iO, y, z), 02 (%, y,z), 0n (x, y, z), 



the expression of that alternation will be 



0! (#, y, z) + 2 0, y, z) . . -f M (>,y,*) = 1; 



the literal symbols x 9 y, z being logical, and relating to the sim- 

 ple events of which the three alternatives are compounded : 

 and, by hypothesis, the expression of the probability that some 

 one of those alternatives will occur is 



0, (#, y, z) + 02 (x, y, z) . . + n (x, y, z), 



* 



x, y, z here denoting the probabilities of the above simple events. 

 Now this expression increases, cceteris paribus, with the increase 

 of the number of the alternatives which are involved, and di- 

 minishes with the diminution of their number ; which is agree- 

 able to the condition stated. 



Furthermore, if we set out from the above hypothetical defi- 

 nition of the measure of probability, we shall be conducted, 

 either by necessary inference or by successive steps of suggestion, 

 which might perhaps be termed necessary, to the received nu- 

 merical definition. We are at once led to recognise unity (1) 

 as the proper numerical measure of certainty. For it is certain 

 that any event x or its contrary 1 - x will occur. The expres- 

 sion of this proposition is 



x + (1 - x) = 1, 



whence, by hypothesis, x+ (1 - x), the measure of the proba- 

 bility of the above proposition, becomes the measure of certainty. 

 But the value of that expression is 1, whatever the particular 

 value of x may be. Unity, or 1, is therefore, on the hypothesis 

 in question, the measure of certainty. 



Let there, in the next place, be n mutually exclusive, but 

 equally possible events, which we will represent by t ly 2 ? t n . 



