728 Miss Wrinch and Dr. H. Jeffreys on some 



We therefore have the theorem : if a selection of p-\-q mem- 

 bers from a class of m a ? s consists of p /3's and q not-/3's, and 

 f(n) the prior probability of there being n ft's and ??i — n not-/3's 

 in the class a. is such that when a; is numerically greater than 

 €,f(n) is never so great that f(n) exp — 2(p-\- q){ar — e 2 ) is 

 comparable with f{mp/(p + q)}, then the probability that 

 njm lies within e oip/(p + q) differs from unity by a quantity 

 of order not greater than exp — 2(p + q)e 2 . Thus, unless the 

 distribution of prior probability among various values of n 

 is very remarkable, its precise form does not produce much 

 effect on the probability that the true value lies within a 

 certain range determined wholly by the constitution of the 

 sample itself. 



It is worthy of note that the range within which it is 

 probable that njm must lie is of length proportional to 

 {p + q)~* ; it does not depend on m to any great extent, but 

 if p and q are very different the range may be much shorter 

 than this. This leads to the result that there is a strong 

 presumption that a large sample is approximately a fair one 

 even if it is small compared with the whole of the class; and 

 that the range within which the fractional composition is as 

 likely as not to lie is much the same however great the 

 whole number of individuals may be. The fact that the error 

 likely to be committed in sampling is, except in extreme 

 cases, limited by the size of the sample itself, may be of some 

 importance in electoral and economic questions. It is also 

 easy to infer from the results obtained that the probability 

 of drawing a j6 at the next trial is not likely to be far from 

 pfip + q), agreeing with sufficient accuracy with the result of 

 the ordinary theory. 



In a recent paper * Mr. C D. Broad has given a suggestive 

 discussion of the problem of inductive inference, in which he 

 adopts the ordinary theory, according to which when q is 

 the probability that all the members are /3's is (p + l)/(«i + 1). 

 This is not necessarily true, for the reasons given above, but 

 this does not affect Mr. Broad's main point, which is that 

 in all ordinary cases the number of observed instances is so 

 small compared with the total number of instances that it is 

 impossible to arrive by this means at any noteworthy prob- 

 ability for a general law. General laws are, however, of 

 various kinds. The type to which Mr. Broad devotes most 

 attention is the statement that " all crows are black, " based 

 on the fact that all observed crows have been black. Now a 

 crow is an object denned by the conjunction of a number of 

 properties, which may or may not include blackness. In the 

 * < Mind,' October 1918, pp. 389-404. 



