the Theory of Probabilities. 461 



bability of the phsenomenon would be the same in both cases. 

 But in one case he woukl know that other causes might have 

 produced it much more easily than accident ; in the other, he 

 would not know this. It is to be observed, however, that in 

 fact, however strong the o priori evidence might be of the 

 absence of any cause but accident, it would be outweighed by the 

 evidence afforded by a certain degree of symmetiy, of the inter- 

 ference of some regulating cause. So that the' surprise could 

 never go beyond a certain point, at which it would be superseded 

 by a conviction that the phsenomenon was not accidental. 



7. I have thought it worth while to say so much upon this 

 question, because it seems to in\olve something more like a real 

 difficulty than any other connected with the subject. The above 

 explanation appears to me to be perfectly satisfactory, and to be 

 equally in accordance with the mathematical theory and with 

 common sense. I am far from su])posing that there is anything 

 original in it, though I am not able to refer to any place where 

 the difficulty is explicitly mentioned and discussed. 



Tlie second question alluded to above is little more than the 

 converse of that which has just been examined. Why do we so 

 easily believe recorded e\'ents, which beforehand would have an 

 enormous a priori improbability ? such, for instance, as any jjar^ 

 ticular unsi/mmetricfit distribution of the counters. So I'ar as 

 general explanation goes, what has been said above will suggest 

 a sufficient answer. But it will hardly be superfluous to notice 

 the mathematical solution. 



8. Let/) be the rt/7?7on probability of an event which a witness 

 has asserted to have happened. And let the a priori probabilities 

 that he would choose to assert it be v on the supposition of its 

 being true, and w on the supposition of its being false. Then 

 after his assertion, the probability that it really happened is 



pv 

 pv-\- {\—p)w' 

 and we see that, however small p may be, the value of this frac- 

 tion may approach indefinitely to unity, provided that v: be much 

 less than v ; that is, in conunon language, provided that the fact 

 of the assertion having been made may be 7niich more easily ac- 

 counted fur by the hyjiotliesis of its truth than of its falsehood. 

 Thus also we see how a priori improbability may even increase 

 a posteriori probability, and vice versa, in cases where the witness, 

 supposing him to lie, would be more likely to choose a ])robable 

 than an im])robable lie ; the smallness of ;; being more than 

 comj)cnsated by the smallness of in, and the converse. 



It would be irrelevant, however interesting, to consider this 

 subject fiu-thcr, because enough has been said to illustrate the 

 real meaning and use of these a priori probabilities, and the way 



