282 BELL SYSTEM TECHNICAL JOURNAL 



the chance of a particular ball being drawn more than once is zero. 

 But when N drawings with replacements are made from a bag con- 

 taining a finite number, M, of balls, we are by no means certain of 

 drawing N different balls; a particular white ball may be drawn several 

 times over and, likewise, a particular black ball may appear more than 

 once. It is not surprising, therefore, that Case 1 of the bag problem 

 does not confirm Bayes' corollary. 



Consider now Case 2, where the drawn balls are not returned to the 

 bag. If k of the total balls are white and the rest black, the probability 

 that a sample of A^ balls from the bag will contain T white and N — T 

 black is 



k\(M - k\ I / M' 

 Tj \N- Tj I \ N 



Hence, if the permissible values 0, 1, 2, 3, • • • M for k are all equally 

 likely a priori, we obtain instead of (7), 



a result independent of any assigned value for T and identical with the 

 result in the corollary to Proposition 8 of the essay. 



Summary 



Bayes' theorem is the answer to a special case of the general problem 

 of causes. The special case postulates that the a priori existence 

 probabilities for the various admissible causes of an observed event are 

 equal. 



In the essay Bayes recommends that his theorem be adopted when- 

 ever we find ourselves confronted with total ignorance as to which one 

 of several possible causes produced an observed event. To justify this 

 recommendation Bayes takes the attitude that: a state of total 

 ignorance regarding the causes of an observed event is equivalent to 

 the same state of total ignorance as to what the result will be if the 

 trial or experiment has not yet been made. This interpretation is a 

 generalization of the fact that in his billiard table problem, the as- 

 sumption of equal likelihood for all possible positions of the line OS, 

 gives equal probabilities for the various possible outcomes of a set of N 

 ball rollings not yet made. 



Laplace, Poincare and Edgeworth ^ have shown that the a priori 

 existence inncXAon iv{x), which appears in the Laplacian generalization 



8 Laplace: "Oeuvres," Vol. 9, p. 470. Poincare: "Calcul des Probabilites," 2d 

 edition, p. 255. Rowley: "F. Y. Edgeworth's Contributions to Mathematical 

 Statistics," pp. 11 and 12. 



