20 Mr. T. K. Abbott on the Probability of 



in half of which instances he affirms white; hence 



chances that white is affirmed on this supposition 



=(i-«)(W)£E|' 



On the whole, therefore, he affirms white in 



, ,.. . m— 2 ,, w , . m— 2 

 «P + (1 ~*)P g^2 + P "^ P "^ 3^3 CaS6S ' 



of which he is right in ap + (l — a)p = ^ cases. Hence the 



v Jr 2m— 2 



probability that wdiite was drawn is 



, ,, i 771 — 2 , M W1 i 771 — 2 



^ + (1 ~^2-^2 + ( 1 -^( 1 -^2^2 

 or putting fl+(l-n) |^ = «, it is 



_ «P 



«p + (l-«)(l-^) 



His credibility in these particular circumstances, therefore, is 



not a, but a. When m is considerable, « nearly = ; f. £. 



<v 



the odds in his favour, instead of a to I — a, are now l + # to 

 1— a, or more than double. 



The result will be the same if we introduce the supposition 

 that the witness intends to deceive, since the motives which 

 usually affect his veracity cannot be supposed to operate uni- 

 formly in favour of the assertion of white when the remaining 

 colours are excluded. Indeed we ought rather to suppose that 

 in the absence of these colours a proportionate number of the 

 motives to deception disappears ; so that if his veracity be v 



and his judgment r, he now affirms white, truly in ap(I—v) _, 



cases, in which, if the other colours were present, he would be 

 induced to affirm some one of them falsely. 



But it is useless to pursue this hypothesis any further. 

 Enough has been said to show that it is altogether unfit to 

 furnish a general formula. It would be quite as reasonable to 

 neglect the probability of the event, and limit ourselves to that 

 of its assertion, as to neglect the latter probability altogether. 

 The general formula already given is independent of any hy- 

 pothesis with respect to the knowledge of the witness, or the 



