BAYES' THEOREM 281 



that T times the ball will rest to the right of OS and that the remaining 

 N — T times it will rest to the left of OS is (as shown in the corollary) 



a result in which T does not appear. In other words, any assigned 

 outcome for the throws is no more, or no less, likely than any other 

 outcome, if a priori all values of x are equally likely. But, wrote 

 Bayes in the scholium, when we say that we have no knowledge 

 whatever a priori regarding the ratio .v, do we not really mean that we 

 are in the dark as to what will be the outcome when we proceed to 

 make N throws? If so, then equat-on (6) justifies the assumption that 

 a priori all values of x are equally likely. 



To clinch his argument it must be shown that the converse of 

 equation (6) is true. That is, it must be shown that, if any outcome 

 of throws not yet made is as likely as any other, then any value of x is a 

 priori as likely as any other. This converse theorem was submitted 

 to Dr. F. H. Murray who obtained an elegant proof based on a theorem 

 of Stieltjes.^ 



In view of Bayes' corollary and his scholium, an analysis of our bag 

 problem with reference to the "equal distribution of our knowledge, or 

 ignorance" is in order. 



Consider again Case 1 where each drawn ball is replaced in the bag 

 before the next drawling is made. 



Assuming each of the {M +1) permissible hypotheses to be a priori 

 equally likely, the probability that A^ drawings, not yet made, will 

 result in T white and N — T black balls is 



P^t,j^(i)i^yi^-^Y-. (7) 



tTo .!/+,! \ T J \M J \ Mj 



Equation (7) is not, in general, independent of T^ so that any one 

 assigned outcome of N drawings is not as likely as any other outcome. 

 This result is disturbing; at first sight it seems to discredit Bayes' 

 scholium. \\'e must, therefore, look into the matter more closely. 



Bayes' problem corresponds to drawings from a bag containing an 

 infinite number of balls. Therefore, even if drawm balls are replaced, 



^Bulletin of the American Mathematical Society, February 1930. 



^ Consider, for example, the case of M = 2. Equation (7) reduces to 



a result which is not independent of T. 



