CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 255 



2. DEFINITION. Two events are said to be independent 

 when the probability of the happening of either of them is 

 unaffected by our expectation of the occurrence or failure of 

 the other. 



From this definition, combined with Principle II., we have 

 the following conclusion : 



V. The probability of the concurrence of two independent 

 events is equal to the product of the separate probabilities of 

 those events. 



For if p be the probability of an event x 9 q that of an eventy 

 regarded as quite independent of #, then is q also the probability 

 that if x occur y will occur. Hence, by Principle II., pq is the 

 probability of the concurrence of x and y. 



Undqr the same circumstances, the probability that x will 

 occur and y not occur will be./> (1 - q). For p is the probability 

 that x will occur, and 1 - q the probability that y will not occur. 

 In like manner (1 - p} (1 - q) will be the probability that both 

 the events fail of occurring. 



3. There exists yet another principle, different in kind from 

 the above, but necessary to the subsequent investigations of this 

 chapter, before proceeding to the explicit statement of which I 

 desire to make one or two preliminary observations. 



I would, in the first place, remark that the distinction be- 

 tween simple and compound events is not one founded in the 

 nature of events themselves, but upon the mode or connexion in 

 which they are presented to the mind. How many separate par- 

 ticulars, for instance, are implied in the terms " To be in health," 

 " To prosper," &c., each of which might still be regarded as 

 expressing a " simple event" ? The prescriptive usages of lan- 

 guage, which have assigned to particular combinations of events 

 single and definite appellations, and have left unnumbered other 

 combinations to be expressed by corresponding combinations of 

 distinct terms or phrases, is essentially arbitrary. When, then, 

 we designate as simple events those which are expressed by a 

 single verb, or by what grammarians term a simple sentence, we 

 do not thereby imply any real simplicity in the events them- 

 selves, but use the term solely with reference to grammatical 

 expression. 



