726 Miss Wrinch and Dr. H. Jeffreys on some 



This gives the solution of the problem in the most general 

 case, but in most cases more concise information, even though 

 it may be only approximate, is desirable. The case ex- 

 clusively considered in the discussions hitherto given is the 

 very simple one where f(n) is the same for all values of n. 

 The ground for this evaluation may be either complete 

 ignorance of the relative number of white balls among the 

 balls in the world, or knowledge that white and other balls 

 have occurred equally frequently in all the ratios possible in 

 this problem. On this hypothesis it can be shown that the 

 probability of a white ball at the (p + q-\-l)th drawing is 

 ( p + l)/(p + q + 2) ; and if q is 0, the probability that all the 

 balls in the bag are white is (p-\-l)/(m+l). 



It is however very rarely, if ever, possible to assume, on the 

 data available before the sample is taken, that/(«) is inde- 

 pendent of n, and cases where it has other values are much 

 more interesting. For instance, we know that there is a 

 strong tendency for similar individuals to be associated, so 

 that the greatest and least values of n are more probable on 

 the initial data than the intermediate ones. Or suppose we 

 are considering balls of another colour, say green. It would 

 be absurd to suggest that a bag is as likely to contain green 

 balls alone as to contain no green balls, for we know that in 

 fact green balls are not nearly so common as balls of all 

 other colours together. On the other hand in these cases it 

 is not usually possible to decompose the propositions, whose 

 probabilities we wish to assess in order to find /(ft), into 

 equally probable alternatives, so that the principle of sufficient 

 reason cannot be applied ; thus though we may be confident 

 that /(ft) lies within certain limits, we cannot say that it has' 

 any particular value. It will, however, be shown that unless 

 the form of this function is something very remarkable the 

 probabilities to be assigned to particular values of n are 

 practically independent of the prior probabilities, depending 

 almost wholly on the composition of the sample taken, pro- 

 vided this is large enough. To show how this comes about 

 we need an approximation to n O p m ~ n C q when p and q are 

 fairly large. This is best obtained by a method analogous 

 to that adopted by Dr. Bromwich *. If r and s are both 

 large and r is large compared with s, formula (2) of 

 Dr. Bromwich's paper yields the approximation 



<? 2 

 lo g { (r +'*)!} = (r + s + J) log r-r + llog27r + £- 



,3 



s 

 -f terms of order -= &c. (2) 



Loc. cit. 



