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



or he is wrong and black is drawn, 



on which supposition the probability of the hypothesis, I. e. 

 the chance that he affirms white out of the m—1 colours which 

 are not drawn, is 



1 



• 



m—1 



The chances in favour of this supposition therefore are 



. 71-1 1 



1— a T « 



n m — 1 



Hence the probability of the event affirmed 



(m — \)a 



(m— l) fl + (l— a)(n— 1) 



If there are several (r) independent witnesses who give the 

 same evidence, then, supposing their credibility to be the same, 

 the probability of the event is 



(m— l) r a r 



(m — l) r a r +(l — a) r (n— 1) 



In the example above given of 10 balls and 3 witnesses each 

 of credibility §, if we suppose only four colours capable of being 

 mentioned, this formula gives as the probability of the event 

 |^ ; that is, the odds in favour of the event are 24 to 1 instead 

 of being only 8 to 9. 



If now we return to the case put by Laplace, it may be asked, 

 is it possible that the circumstance of the witness knowing or not 

 knowing that there are only two colours in the urn can make 

 such an enormous difference in the credibility of his testi- 

 mony? I answer no; and for this reason, that a does not 

 represent the same quantity in both cases. This will be at once 

 obvious from the following consideration. When there are 100 

 numbered balls, a man whose announcements are made altogether 

 at random will be right only once in a hundred times ; but a man 

 whose credibility is \, i. e. who is right fifty times in a hundred, 

 is a good witness. Two such witnesses agreeing would be equi- 

 valent to one whose credibility is -f^. But if there are fifty 

 black and fifty white balls, so that there are only two pos- 

 sibilities to choose from, and the chances of these are equal, the 

 random speaker will be right fifty times in a hundred. In this 

 case, therefore, the credibility represented by \ is no credibility 



