Testimony and Arguments. 17 



is true), then the three together do not make it an even chance 

 that white was drawn. The odds are still 9 to 8 against it ; and, 

 as before noticed, if the testimony of these is supported by a 

 number of independent witnesses whose average credibility is 

 only J-, it does not become a whit more probable. Common 

 sense shows that this is to allow too much to the extraordinary 

 character of the event, and that some condition must be involved 

 in the calculation which does not apply to the events of ordi- 

 nary life. 



One such condition was introduced by the assumption 

 that when the witness is mistaken or deceived, there is no 

 diversity of error possible. There are only two colours, and he 

 knows this ; therefore if black is drawn and he is wrong, it can 

 only be by testifying to white. In order to apply such a for- 

 mula to the evidence for an extraordinary event in general, we 

 must assume that, supposing it not to have occurred, any error 

 on the part of the witnesses must have led to its being reported. 

 Of course whenever this can be shown, the circumstance detracts 

 immensely from the weight of the testimony ; but it is very far 

 indeed from being generally the case. Before examining the 

 problem more generally, it is worth while to show that Laplace's 

 formula applies only to the particular case which he has selected ; 

 and that the extension of it by subsequent writers, who substitute 



p (the antecedent probability) for -, is fallacious, even when we 



frame our conditions in perfect analogy with those of Laplace. 

 Thus suppose, besides the 99 black and one white ball, we put 

 into the urn 99 blue and one yellow, 99 red and one green. The 

 drawing of white is exactly as extraordinary as before, and the 

 chances against it greater, viz. §§f . Nevertheless when it is 

 affirmed its probability becomes (the witness's credibility being |) 



5 25 



5 + 2|9 324 



or more than half as much again as in the former case. We 

 must therefore investigate the problem under more general con- 

 ditions. 



Suppose then that the witness A docs not know anything 

 about the colours or number of the balls in the urn ; his affir- 

 mations, therefore, may range over all possible colours, say 

 m. Now if he states that the ball drawn was white, we have the 

 following cases : — 



Hypothesis. — A affirms white. Either he is right and white 

 is drawn, a 



n ; 

 Phil Mag % S, 4. Vol, 28. No. 186. July 1864. C 



