Aspects of the Theory of Probability. 727 



Put n = ^- ; n=n + mx (3) 



p + q 



Then 

 log ( n C p m ~ n G q ) = log { (rc + mx) ! } + log { (m — n — mx) ! } 

 -logp! -logy! 

 — log {(n —p + mx) !} - log {(m-n —q—mx) !}, (4) 



which gives on substituting the above approximation, pro- 

 vided mx is not so great as to invalidate it, an expression 

 that simplifies to 



(p + q) log — — + (m-p — q + 1) log 1 log (pq) 



-logg^y fr +?)*"»■ . (5 ) 



* pq(m-p-q) 



Hence the function considered is a maximum for n = n , and 

 its values for other values of n are distributed about this 

 according to the Gauss law. 



The given sample is said to be a fair one i? p/(p-\-q) is 

 equal to the ratio of the true number of white balls to the 

 whole number of balls in the bag. The deviation from 

 fairness is therefore represented by x. Substituting the 

 approximation (5) in the formula (1), we find by summation 

 that the probability that x lies between +e is 



2' f(n) exp 



1 (p + qYmx 2 



*PV( m -P-9) 9 . . . (6 ) 



S/(»)exp.-i-i£+«^ 



pq(m — p — q) 



where in the denominator the summation covers all values of 

 n and in the numerator all values between ?n( — e ) and 



\p + q J 



m 



i — -— -f e j. Now the coefficient of x % in the exponent 



is always numerically greater than 2 {p + q)> If then 

 2%(p + q)h is greater than h, the exponential is less than e~ h '\ 

 which is very small even when h is not remarkably small. 

 Outside of the range the exponential factor is even smaller, 

 and unless /(n) is so great that its greatness can counteract 

 the smallness or! the exponential factor, the contribution to 

 the denominator from the values of x not between +e is 

 small. Thus the numerator and denominator are nearly 

 equal and the probability that x lies between + e is nearly 1. 



