Aspects of the Theory of Probability. 725 



show that the conclusion has a certain probability relative to 

 the premises ; an inference of this kind may be called a 

 "probability inference/' The establishment by this means 

 of a high probability in favour of the conclusion relative to 

 the premises is often as useful as the inference that does not 

 involve the notion of probability. The course thus indicated 

 is always followed in empirical generalization, for in such a 

 generalization it is never possible to establish the certainty 

 of the conclusion from the data. The principles employed in 

 such inference are therefore of extreme importance ; but as 

 yet they are not well understood. 



Detailed treatment is most applicable to the type of 

 probability inference known as sampling induction, and 

 numerous discussions of this have been given, but even here 

 various errors seem to have survived. The problems capable 

 of solution by this method are analogous to the following. 

 Suppose that a bag contains m balls, an unknown number of 

 which are white. Of these p + q have been drawn and not 

 replaced ; p of them have been white and q not white. 

 "What is the probability that the number of white balls in the 

 bag is n ? 



It is assumed that the balls are indistinguishable before 

 being drawn, so that at any stage any individual ball is as 

 likely to be drawn as any other. Let /(«) be the prior 

 probability of any particular number of white balls. If n 

 were the true number of white balls in the bag the probability 

 that p white balls and q others would be picked in p + q 



trials would be — ~ — . It follows that the prior 



yp+q 

 probability of a particular pair of values of p and q for a 



nH m—nH 



given n is f(n) — ^ q . Hence, by the law of inverse 



probability, which follows easily from the proposition 



the probabilities on the data of particular numbers of white 

 balls are in the ratio of the probabilities of the actual values 

 of p and q for these numbers of white balls ; thus we find 

 that the probability that any particular value of n is the true 

 number of white balls in the bag, given the composition of 

 the sample, is 



/(w) Up C q -i~Z n f(n) Op G qi . . (1) 



where the summation is to be extended to all possible values 

 of n. 



