On the Mathematical Theory of Probabilities. 511 



hypothesis can be shown to involve; and its consequent rejec- 

 tion clears the way, at least, for the establishment of the 

 newer mode of stating the case. 



In a volume recently published of c Studies in Logic/ Mr. 

 C. S. Peirce raises this question of inconsistency, and shows 

 in a particular case that, if all frequencies of the event Y are 

 equally probable, the frequencies of the event which consists in 

 a Y following a Y or an N following an N are not equally 

 probable. The hypothesis is thus shown to be inconsistent, 

 since it cannot simultaneously be applied to these two unknown 

 events of Y and Y repeated. The demonstration of inconsist- 

 ency which follows is suggested by Mr. Peirce^s, but differs 

 from his in its generality both of method and result, and also, 

 I venture to think, in the strictness with which it keeps 

 within the limits of the hypothesis in question. Mr. Peirce, 

 it appears to me, uses Boole's conception in proving the incon- 

 sistency of the rival conception. While following his therefore 

 in essentials, my proof differs altogether in details. 



Let x denote the occurrence of an event, 

 x „ its non-occurrence ; 



and, for brevity, let 



x n denote the occurrence of the event in n cases out of 



n possible cases, 

 x n ~ l x denote the occurrence of the event in n— 1 cases 



out of n cases, 

 x n ~ 2 x 2 denote the occurrence of the event in n — 2 cases 

 out of n cases, 

 and so on. 



It is required to investigate the probabilities of the various 

 relative frequencies of the event which consists in x following 

 x, or x x, i. e. the probabilities of all the conceivable proba- 

 bilities of this event. 



If x n occurs, xx occurs n — 1 times out of n — 1. 

 If x n ~ l x occurs, two cases have to be considered. Either 

 x is last in the series, or it intervenes between two #'s. In the 

 first case x x occurs ?i — 2 times] in the second case it occurs 

 7i — 3 times. Now our assumption gives us no clue to de- 

 termining the respective probabilities of these two cases. A 

 new assumption seems necessary, and the suggestion of Boole's 

 would, I think, naturally arise here*. We resist the sugges- 

 tion, however, and, keeping strictly within the prescribed 

 limits, are driven upon indeterminateness. Call the proba- 

 bility that x is last in the series X 1; and the probability that x 



* By its implicit use Mr. Peirce arrives at a determinate instead of an 

 indeterminate result. 



2M2 



