Testimony and Arguments. 15 



of the former are ^j and as there are 7 balls not of this 

 colour, of which 2 are white, and the witness knows this, the 

 chances of white being falsely named in this case = ^ f = 3V 

 Similarly, the chances of a false assertion of white when red is 

 drawn =^ f = 3^- Total false assertions of white ==^%, to be 

 multiplied by the chance that A speaks falsely = ^, 

 Hence, finally, the probability of A J s assertion is 



- - * _ 7 



In general, on this supposition, if there are n cases whose 

 probabilities are^? 1} jo 2 , &c, the sum of these being 2, the chances 

 of a false assertion of p x are 





Call the quantity within the brackets C ; then if the credibility 

 of the witness be a, and he asserts the event »,, the probability 

 that he is right is 



a 

 a + (l-a)C' 

 If we suppose that the witness knows the colours of the balls, 

 but not the number Of each colour, the case is different. Sup- 

 pose, with the same numbers, he affirms white. Either he is 

 right, and it is drawn 



— 1 



— 6* 



Or he is wrong, and selects white from the two colours not drawn, 



= 6 10 2 = 15* 



Resulting probability of white 



_ i 5 



In general, if there be n cases p l} p q , &c. as above, and the wit- 

 ness, knowing only the number of possible cases, asserts the 

 event p v the probability that he is right is 



(*- l)qgi_ , 



Thus it is clear that on either hypothesis the case assumed by 

 Laplace is not general. The condition involved in it, stated 

 generally, will be seen to be, that, if the antecedent probability 

 of the event named be p, the chances of its being falsely named 

 when it has not occurred are 



P 

 1-p 



