BA VES' TIIF.ORKM 279 



Each rolling of the ball was executed in such a manner that the 

 probability of the ball coming to rest to the right of OS is given by the 

 unknown ratio of the distance OA to the length BA of the table; 

 likewise, the probability of the ball stopping to the left of OS is given 

 by the ratio of the distance BO to the length BA . 



Set X = OA/BA, 1 - x = BO/BA. 



The only difference between this problem and the bag of balls 

 problem is that now the possible values of x are not restricted to the 

 finite set 0/M, 1/M, 2/M, • • • (M - 1)1 M, M/M; in the table problem 

 X may have had any value whatever between the limits and 1. 

 Therefore equation (3) will answer the question asked provided we 

 substitute definite integrals in place of the finite summations. This 

 substitution gives us, for the desired a posteriori probability that x had 

 a value between ;Vi and X2, the formula 



I iv(x)x'^{l — x)''^~''^dx 

 P(xu X,) = -^^ (4) 



Jo 



Equation (4) is useless until the form of the a priori existence function 

 w{x) is specified; this depends on the way in which the line OS was 

 drawn. Bayes assumed that the line OS, of unknown distance from 

 AD, was drawn through the point of rest corresponding to a preliminary 

 roll of the ball. This amounts to postulating that all values of x, 

 between and 1, were a priori equally likely. In other words, with 

 Bayes, the a priori existence function iv{x) was a constant which, 

 therefore, did not have to be taken into consideration.^ Thus, instead 

 of equation (4), Bayes gave the equivalent of the following restricted 

 formula: 



r \r''(l - xY'^dx 



P{xu X.) = 7i ; (5) 



I .r^(l - xy-'^dx 



I say "the equivalent of" (5) because in Bayes' day definite integrals 

 were expressed in terms of corresponding areas. 



Equation (5) constitutes Proposition 9 of the essay, but is usually 

 referred to as Bayes' theorem. 



3 The existence function u^.v) does not appear eitlier exi^licitly or implicitly any- 

 where in Bayes' essay. This fact raises the question as to whether or not Bayes had 

 any notion of the general problem of causes. 



