SECT. 13.] Inverse Probability. 183 



having been drawn from a bag which only contained one of 

 these white balls. Of these there is, out of the 55 drawings, 

 but one. Accordingly the required chance is ^. That is to 

 say, the white ball will have been drawn from the bag con 

 taining only that one white, once in 55 times. 



13. Now, with the exception of the passage in italics, 

 the process here is precisely the same as in the other exam 

 ple ; it is somewhat longer only because we are not able to 

 appeal immediately to experience, but are forced to try to 

 deduce what the result will be, though the validity of this 

 deduction itself rests, of course, ultimately upon experience. 

 But the above passage is a very important one. Tt is scarcely 

 necessary to point out how arbitrary it is. 



For is the supposition, that the different specified kinds 

 of bags are equally likely, the most reasonable supposition 

 under the circumstances in question ? One man may think 

 it is, another may take a contrary view. In fact in an excel 

 lent manual 1 upon the subject a totally different supposition 

 is made, at any rate in one example ; it is taken for granted 

 in that instance, not that every possible number of black and 

 white balls respectively is equally likely, but that every 

 possible way of getting each number is equally likely, whence 

 it follows that bags with an intermediate number of black 

 and white balls are far more likely than those with an ex 

 treme number of either. On this supposition five black 

 and five white being obtainable in 252 ways against the 

 ten ways of obtaining one white and nine black, it fol 

 lows that the chance that we have drawn from a bag of 

 the latter description is much less than on the hypothesis 

 first made. The chance, in fact, becomes now ^ instead 

 of ^. In the one case each distinct result is considered 



1 Whitworth s Choice and Chance, Ed. n., p. 123. See also Boole s 

 Laws of Thoiujht, p. 370. 



