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



Now let us sec how Laplace treats the case of a witness attest- 

 ing a fact of given antecedent probability. 



Suppose there are 100 balls, of which one is white and the 

 rest black. The witness whose credibility is |- asserts that the 

 ball drawn was white. The cases are : — 



Hypothesis, — A affirms white. Either he is right = |, and 

 white was drawn = t 1q; 



chances in favour of this supposition = gg^. 



Or he is wrong = \, and black was drawn = -f^ ; 



chances in favour of this supposition = ^5. 



For on this supposition the hypothesis is certain, inasmuch as A, 

 if he is wrong, must bear witness to white. Hence the total 

 probability of white is 



5 5 



5 + 99 ""104* 



In general if the whole number of balls be n, of which one only 

 is white, the rest being black, and the witness whose credibility 

 is a asserts that white was drawn, the probability of this event 

 is evidently 



c+(l— a)(n — 1) 

 The probability that the assertion is not true is 



(*-l)(l-a) . 

 a+(n— I)(i — a) 



H n is very large, this will not differ much from unity, i. e. 

 certainty, if we suppose even a slight chance of mistake or decep- 

 tion on the part of the witness. Now this is the case of an extra- 

 ordinary event ; and the result, adds Laplace, confirms the judg- 

 ment of common sense, that such an event requires far stronger 

 evidence to prove it than an ordinary one. 



On the same conditions, if several (say r) witnesses join in 

 attesting the drawing of white, then if their credibility is the 

 same, the resulting probability is 



a r +(l — a) r (n — l) 



This corresponds with the formula quoted at the beginning of 

 this paper, and the same consideration shows that it is not appli- 

 cable to such cases as are met with in ordinary experience. Sup- 

 pose, for instance, the number of balls is 10, and three indepen- 

 dent witnesses testify that white was drawn. If the credibility 

 of each is § (i. e., it is 2 to 1 that his unsupported testimony 



