CHAP. XVII.] GENERAL METHOD IN PROBABILITIES. 259 



sible combinations of the two simple events, x and#, of which the 

 respective probabilities are p and q. The primary combinations 

 of those events (V. 11), and their corresponding probabilities, are 

 as follows : 



EVENTS. PROBABILITIES. 



xy. Concurrence of a? and y, pq. 



x (1 - y) 9 Occurrence of x without y, p (1 - q). 



(1 - x)y, Occurrence of y without x, (1 - p) q. 



(1 - x) (1 - #), Conjoint failure of x and y, (1 - p) (1 - q). 



We see that in these cases the probability of the compound event 

 represented by a constituent is the same function of p and q as 

 the logical expression of that event is of x and y ; and it is obvious 

 that this remark applies, whatever may be the number of the 

 simple events whose probabilities are given, and whose joint ex- 

 istence or failure is involved in the compound event of which we 

 seek the probability. 



Consider, in the second place, any disjunctive combination of 

 the above constituents. The compound event, expressed in or- 

 dinary language as the occurrence of " either the event x without 

 the event y, or the event y without the event x" is symbolically 

 expressed in the form x (1 - y) + y (1 - or), and its probability, 

 determined by Principles iv. and v., is p (1 - q) + q (1 - p). The 

 latter of these expressions is the same function of p and q as the 

 former is of x and y . And it is obvious that this is also a par- 

 ticular illustration of a general rule. The events which are ex- 

 pressed by any two or more constituents are mutually exclusive. 

 The only possible combination of them is a disjunctive one, ex- 

 pressed in ordinary language by the conjunction 0r, in the lan- 

 guage of symbolical logic by the sign +. Now the probability of 

 the occurrence of some one out of a set of mutually exclusive 

 events is the sum of their separate probabilities, and is expressed 

 by connecting the expressions for those separate probabilities by 

 the sign +. Thus the law above exemplified is seen to be general. 

 The probability of any unconditioned event Fwill be found by 

 changing in V the symbols x, y, z, . . into /?, q, r, . . 



8. Again, by Principle in., the probability that if the event 

 V occur, the event V will occur with it, is expressed by a frac- 



s2 



