Aspects of the Theory of Probability. Ill 



evidence is for the existence of such a limit. Suppose, for 

 instance, that the probability of -v/r given <f> is J. Then the 

 numbers of both ty's and not-^'s are infinite, and selections 

 of (j) } s may therefore be made so that the ratio of i/r's to all 

 <£>'s will tend to any limit whatever between and 1; it may 

 even tend to no limit at all. If, for instance, every time a 

 "yjr occurs we write 1, and every time a not-ijr occurs we write 

 0, m/n will be the mean of the first n terms of the series thus 

 obtained. If then they occur in such an order as to give the 

 series 



1011000011111111 (1) 



where the number of digits in any block of similar digits 

 after the first is equal to the total number of digits that 

 have occurred previously, let us consider the r-th block, 

 starting at the (2 r ~ 2 + l)th figure. If r is even, the number 

 of l's that have already occurred is 



1 + 2 + 8+ ...+2'- 3 = i(2 J - 1 + l), . . . (2) 

 so that 



mjn when n = 2 r ~ 2 is i(2 + 2"^- 2) ). . . . (3) 



The r-th block consists of 2 r ~ 2 zeros, and at the end of it 

 mjn has fallen to ^(l + 2" (r_1) ). In the next block it rises 

 again to ^-(2 + 2~ (r_1) ). Thus, however great r may be, we 

 can find values of n greater than 2 r ~ 2 such that mjn is greater 

 than f, and others such that m/n is less than o+e, however 

 small e may be. Thus m/n tends to no limit whatever. The 

 notion that all series picked from the class of entities with 

 the property (/> will give series of values of m/n tending to 

 the same limit is therefore incorrect, unless some further 

 criterion be introduced to exclude all those that do not behave 

 in this way, whose number is infinite ; and the task will not 

 be an easy one, for any criterion based on the mode of 

 occurrence of long runs of ijr's or not-^'s is liable to be 

 found invalid in instances occurring in practice. 



The origin of the idea that such a limit must exist may be 

 considered at this stage, as it involves a theoretical point that 

 may be of importance in future developments of the subject. 

 The belief was based on a well-known theorem of James 

 Bernoulli, a proof of which, based on Stirling's approximation 

 to w! for large values of n, is given by Laplace*. This 

 theorem answers the following question : If the prior pro- 

 bability of a ^ be r, however many ty's and note's have 



* Loc. cit. pp. 275 et seqq. A proof based on the same principle, but 

 more elegant and easily applied, is given by Bromwich, Phil. Mag. 

 August 1919, pp. 231-235. 



Phil. Mag. S. 6. Vol. 38. No. 228. Dec. 1919. 3 D 



