180 Inverse Probability. [CHAP. vn. 



inference can be immediately derived. All other distinctions 

 are merely those of arrangement or management. 



But although the distinction is not by any means fun 

 damental, it is nevertheless true that the practical treatment 

 of such problems as those principally occurring in Inverse 

 Probability, does correspond to a very serious source^ of 

 ambiguity and perplexity. The arbitrary assumptions which 

 appear in Direct Probability are not by any means serious ; 

 but those which invade us in a large proportion of the prob 

 lems offered by Inverse Probability are both serious and 

 inevitable. 



10. This will be best seen by the examination of 

 special examples; as any, however simple, will serve our 

 purpose, let us take the two following : 



(1) A ball is drawn from a bag containing nine black 

 balls and one white: what is the chance of its being the 



white ball ? 



(2) A ball is drawn from a bag containing ten balls, and 

 is found to be white ; what is the chance of there having 

 been but that one white ball in the bag ? 



The class of which the first example is a simple instance 

 has been already abundantly discussed. The interpretation 

 of it is as follows : If balls be continually drawn and re 

 placed, the proportion of white ones to the whole number 

 drawn will tend towards the fraction T V The contemplated 

 action is a single one, but we view it as one of the above 

 series ; at least our opinion is formed upon that assumption. 

 We conclude that we are going to take one of a series of 

 events which may appear individually fortuitous, but in 

 which, in the long run, those of a given kind are one-tenth of 

 the whole ; this kind (white) is then singled out by anticipa 

 tion. By stating that its chance is T ^, we merely mean to 

 assert this physical fact, together with such other mental 



