246 Lord Rayleigh 071 James BernouilU's 



on a variety of series in games of chance and on biostatis- 

 tical data, — a small change in a high moment makes a large 

 change in n. Accordingly we are liable to form quite 

 erroneous impressions of the nature of the hypergeometrical 

 series, and even to reach impossible values for p, g, and r 1 

 which are determined through n. Thus the problem, which 

 is practically an important one, as enabling us to test the 

 sufficiency of the usual hypothesis, n = c© , of the theory 

 of errors, i. e. to test the " independence or interdependence 

 of contributory causes," is seen to admit of a solution, but 

 one which is hardly likely to be of much service unless in the 

 case to which it is applied a very large amount of data is 

 available. 



XVI. On James Bernoulli? s Theorem in Probabilities. 

 By Lord Rayleigh, F.R.S* 



IF p denote the probability of an event, then the probability 

 that in jjl trials the event will happen m times and fail n 

 times is equal to a certain term in the expansion of (p+qY> 

 namely, 



■A^-.P^", (i) 



where p+q = l, m + n=r/ji. 



" Now it is known from Algebra that if m and n vary 

 subject to the condition that m-f n is constant, the greatest 

 value of the above term is when m/n is as nearly as possible 

 equal to p/q, so that m and n are as nearly as possible equal 

 to fip and fxq respectively. We say as nearly as possible, 

 because /xp is not necessarily an integer, while m is. We 

 may denote Ihe value of m by pp + z, where z is some proper 

 fraction, positive or negative ; and then n = pq—z" 



The rth term, counting onwards, in the expansion of 

 {p+qY after (1) is 



**! 



m 



? , t p m ~ r q n+r - > . . (2) 



r I n -j- r ! v ' 



The approximate value of (2) when m and n are large 

 numbers may be obtained with the aid of Stirling's theorem, 

 viz. 



^! =/ ^*-^27r)£L+ JL + ...J., . (3 ) 



* Communicated bv the Author, 



