718 Miss Wrinch. and Dr. H. Jeffreys on some 



been chosen already, what is the probability that when n <£'s 

 have been selected mjn will lie between r — e and r + e? It 

 is shown that the probability of any particular value of m is 



(l-r) n - m , .... (4) 



m ! (11 — m) ! 

 and when Laplace approximates according to the formula 



n!=n»,-V(2 7 rn)(l+ I l 7( + ^+--.) . (5) 



he shows that this is a maximum when m is rn } which is not 

 necessarily an integer, and if mjn is equal to »* + f. he gives 

 a formula which reduces to 



{ 2«r(l-r) Y e ~^ l - T) {^0{^)}dt . (6) 



The probability that £ does not lie between + e is then 



+ |tt-^-"rw-,)/ e 2_- lr ^~^ \\ . (7) 



Now, no matter how small e and 77 may be, it is always 

 possible to choose n large enough to make this less than v ; 

 accordingly, by making n great enough we can make the 

 probability that mjn differs from r by more than any quantity 

 assigned beforehand as small as we please. This is Bernoulli's 

 theorem. 



This does not, however, give the probability that mjn 

 will tend to a limit as n tends to infinity. For if e be a small 

 quantity fixed beforehand, the necessary and sufficient con- 

 dition that mjn tend to a r as a limit is that a value of n Q can 

 always be found such that for all values of n greater than n , 

 m/n — r shall be less numerically than e. Now if x be great 

 enouo;h to make e~ x ' small, we have the relation 



1-Erf £ = 



x </> 



{ l +°&)\ ■ ■ ■ m 



and the probability that mjn does not lie between + e is 

 therefore, when n is great enough, 



2 



'•••'V( 2 ^{' + <^)}- • <»» 



