Testimony and Arguments. 19 



at all, and two such witnesses are accordingly of no more value 

 than one ; so that the same degree of credibility is represented 

 in the one case by yoo> anc ^ ^ n tne °th er by \ • ^he credibility, 

 therefore, which was represented by J in the first case ought to 

 be represented by a greater fraction in the second ; the witness 

 has not lost all his credibility because the event itself has become 

 more probable. It is clear, then, that a represents not the 

 antecedent credibility of the witness, but the chances of his 

 announcement being right under the particular conditions sup- 

 posed. 



To consider the matter practically : — the chances of the witness 

 being mistaken in the colour of a black ball are the sum of the 

 chances of his taking it for white, for yellow, for red, and the rest 

 of the m colours. Now if it is known that there are only two 

 colours in the urn, he is excluded from m — 2 of the m — 1 possible 

 mistakes, but we have no reason to suppose that when he is dis- 

 posed to mistake black for red for example, he has no choice 

 but to affirm white if he knows that red is not present. If we 

 know nothing further about the case, we must suppose that in 

 all such instances (i. e. where he mistakes black or white for a 

 colour known not to be present) he has no reason for choosing 

 either black or white, and therefore divides his assertions equally 

 between them. We have then the following cases : — 



Hypothesis. — He affirms white, the probability of which is p, 

 and his credibility a. 



Case 1. He believes that white is drawn, and it is so; 



chances in favour of this supposition = ap. 



Case 2. White is drawn, and he mistakes it for some colour 

 known to be absent ; 



chances = p z, (1—a). 



In half of these instances he affirms black, and the other half 



white; hence 



m— 2 

 chances that he affirms white on this supposition = (1 — a)p- ^. 



Case 3. Black is drawn, and he mistakes it for white ; 



1 



chances = (1 — a) (1 — p) 



m—\ 



Case 4. Black is drawn, and he mistakes it for an absent 

 colour, 



m— 2,, » 



(i-ztera-*)* 



C2 



