Aspects of the Theory of Probability. 719 



The'probability that any value of m/n for n>n lies outside 

 these limits is therefore not greater than the sum of these 

 expressions for all values of n from n Q -+ 1 to infinity. We 



see that this is less than 



2 nra-r) e . ;io ^. (1 _, r x o/MMj 1 



e V w 7r L V V /J 



{ 1 + e - eV2r(l-r) + e - 2eV2r(1 -r) +...}. (10) 



The sum of the series is finite and independent of n ; 

 hence we see that % can always be chosen so as to make the 

 probability that, for all values of n greater than n , the value 

 of m/n will lie between r±e, differ from unity by as small a 

 quantity as we like. 



The proposition required for the validity of "Venn's theory 

 is : n can always be chosen so as to make the probability 

 that, for all values of n greater than n , the value of m/n will 

 lie between r±e, exactly equal to unity. 



These two propositions bear a close resemblance to each 

 other, but they are not equivalent. In fact, in consequence 

 of the existence of modes of selection for which m/n does not 

 tend to r as a limit, we know that the second proposition 

 must be false. The first, on the other hand, has just been 

 proved true ; but it does not even establish a high probability 

 for the proposition that m/n tends to r as a limit in any 

 particular case. For it has been shown only that a certain 

 result will have a very high probability when a single value 

 of e has been assigned ; but there is no reason to infer from 

 this that the probability is high that it will hold for all values 

 of e whatever, which would have to be true it" m/n were to 

 tend to a limit. The difficulty is somewhat similar to that in 

 the theory of infinite series, wiih regard to series that " con- 

 verge with infinite slowness.'" 



II. The Mathematical Theory of Probability. 



An essential assumption in order that analytical methods 

 may be applicable to the theory of probability must now be 

 stated, namely, that a correspondence can be established 

 between positive real numbers and the propositions to which 

 the fundamental notion of probability is applicable (relative 

 in.each case to the appropriate data) which shall have the 

 following properties. 



3D2 



