Testimony and Arguments, 13 



Or he is wrong, 



and the event has not occurred, 

 ,\ the chance of this coincidence is 



1 — a L—p. 



The sum of these gives the common denominator ; and hence 

 the resulting probability, P, that the event has actually hap- 

 pened is 



P = ap 



ap + l—a l—p 



Now let there be two or more witnesses who agree in the same 

 assertion, their respective credibility being a v a 2 , &c. Then 

 either they are together right, and the event has occurred, the 

 chance of which is 



a, # 2 . . . a n p ; 



or they are together wrong, and the event has not occurred, the 

 chance of which is 



1— «, 1 — # 2 ... 1 — a n l—p. 

 Hence the resulting probability of the event is 



«i ff 2 • • • a nP 



P = 



a ] a 2 ... a n p + \—a l 1 — a 2 . .. 1— j 



From the first formula it follows that, if « = \, P=jo ; that is to 

 say, if this witness attests an event, of the chances of which we 

 know nothing, his affirmation is enough to make the odds on it 

 even ; but if we have any other means of estimating the chances, 

 his testimony goes for nothing. This is certainly not practically 

 true. But the second formula leads to a result more obviously 

 absurd. Suppose a l = a 2 = . . . =a n = £; then in this case again 

 V=p; that is to say, if any number of witnesses, supposed 

 to be wholly independent, give coincident testimony, it adds 

 nothing to the probability of the event, unless each separately is 

 sufficiently credible to turn the odds in its favour. If the odds 

 are against the truth of each witness separately, then the greater 

 their number the less credit is due to their testimony. The 

 weight usually attributed to the coincidence of independent wit- 

 nesses is inconsistent with the formula, according to which 

 agreement in truth is just as unlikely as agreement in falsehood. 

 A calculation which yields such results as these must be erro- 

 neous, or imply conditions very different from those of ordinary 



