412 Credibility of Extraordinary Stories. [CHAP. xvn. 



say 10,000 ; that is, suppose that 10,000 balls having been 

 successively drawn out of this bag, or bags of exactly the 

 same kind, he makes his report in each case. His 10,000 

 statements being taken as a fair sample of his general ave 

 rage, we shall find, by supposition, that 9 out of every 10 of 

 them are true and the remaining one false. What will be 

 the nature of these false statements ? Under the circum 

 stances in question, he having only one way of going wrong, 

 the answer is easy. In the 10,000 drawings the white ball 

 would come out 10 times, and therefore be rightly asserted 

 9 times, whilst on the one of these occasions on which he 

 goes wrong he has nothing to say but black. So with the 

 9990 occasions on which black is drawn; he is right and 

 says black on 8991 of them, and is wrong and therefore says 

 white on 999 of them. On the whole, therefore, we conclude 

 that out of every 1008 times on which he says that white is 

 drawn he is wrong 999 times and right only 9 times. That 

 is, his special veracity, as we may term it, for cases of this 

 description, has been reduced from -^ to TiJ 9 7 g. As it would 

 commonly be expressed, the latter fraction represents the 

 chance that this particular statement of his is true *. 



1 The generalized algebraical form be presently made, the reader will 



of this result is as follows. Let p notice that on making either of these 



be the a priori probability of an expressions =p, we obtain in each 



event, and x be the credibility of the case x = \. That is, a witness whose 



witness. Then, if he asserts that veracity =\ leaves the a priori prob- 



the event happened, the probability ability of an event (of this kind) un- 



that it really did happen is affected. 



If, on the other hand, we make 



ex P ressions e( l ual to x and 



1 - x respectively, we obtain in each 

 whilst if he asserts that it did not case p = %. That is, when an event 

 happen the probability that it did (of this kind) is as likely to happen 

 happen is j? (1 -#) as not, the ordinary veracity of the 



p(l - x) + (l-p)x witness in respect of it remains un- 



In illustration of some remarks to affected. 



