14 M. T. K. Abbott on the Probability of 



experience. In fact we have taken no account of the datum 

 that error is manifold, while truth is one. 



Laplace analyzes the case of single witnesses as follows. His 

 method generally is, having stated the chances of every possible 

 case, to multiply each by the chances of the hypothesis in that par- 

 ticular case. Thus, suppose an urn contains 100 balls numbered 

 consecutively, and one is drawn. The witness A, who knows 

 how many balls there are, affirms that the ball drawn was No. 25. 

 Let us suppose his general credibility to be J ; we have the fol- 

 lowing cases : — 



Hypothesis. — A affirms No. 25. 



Either he is right = J, and it was drawn = j-J-q. In this 

 case the hypothesis is certain, therefore the chances of this 

 supposition ^To^tJo- 



Or he is wrong = J, and No. 25 was not drawn, = 1 9 ^ ) ; but 

 this must now be multiplied by the chance of the hypothesis in 

 this case, i. e. the chance that when A is wrong, his false testi- 

 mony will be borne to No. 25. There being 99 balls not drawn, 

 any of which might be asserted falsely, this chance is therefore ^, 

 and the total chances of this last supposition =q^qq 9*9 = 600' 



The chances of the two suppositions respectively are therefore 

 T20 an ^ 6 00 > ano - as these exhaust all possible cases, the total 



-J— 5 



probability of the event affirmed is . 12 ° = -, i. e. it is 



T2 +.600 b 



the same as the witness's general credibility. The erroneous 

 formula first quoted would have given as the resulting proba- 

 bility —rib = _£_. 



120^.600 1KJ * 

 Laplace, it will be seen, has taken account of the diversity of 

 error ; yet his conditions do not correspond perfectly with those 

 of actual experience. In fact the resulting probability in the 

 case supposed was found to depend solely on the credibility of 

 the witness, not at all on the antecedent probability of the event. 

 This is a consequence of the supposition that the witness knows 

 the number of balls in the urn ; so that the number of possible 

 errors is 99, and the odds against any particular ball are also 

 99 to 1. In fact the chance of any ball being named at random 

 is the same as the chance of its being drawn. But there is a 

 further condition implied in the mode of stating the problem. 

 Let it be supposed that there are 2 white balls, 3 yellow, 

 5 red, in all 10. The witness asserts that white is drawn, his 

 credibility being still supposed =|. The chances in favour of 

 his assertion are 



§ . At = J - 



6 ' 10 6* 



If he is wrong, either yellow was drawn or red. The chances 



