118 Mr Campbell, The study of discontinuous phenomena. 



of the phenomena that we propose to investigate is that they are 

 discontinuous, that is to say, that they are to be regarded as made 

 up of a finite number of events and not of an infinite number: in 

 some eases to which we may have to apply our theory it is by no 

 means certain tliat the number is even very hxrge. By using the 

 integral calculus we assume in our mathematical considerations 

 the proposition which we deny in our physical considerations. 

 Accordingly we must confine ourselves throughout to finite 

 quantities. 



Consider a series of s trials in each of which one of the 

 two mutually exclusive events A and B must happen. Let the 

 probability that A happens be p and the probability that B 

 happens be q. Then it can be shown easily that if A happens 

 m times and B m — s times, the most probable value of in is ps: 

 and further that the probability that A happens ps — x times is 



s\ 



V 



ps-x f^qs+X 



{ps-:v)l{qs + :vy. 



We shall require the mean value of the 'deviation' w for 

 a very large number (<r) of sets of s trials. There are three 

 chief forms of mean value, ;r, x, and \x\. Remembering the 

 identities 



c{p + qy-^ .q = S nA„p'-" q" 



n 



and c{c — l){p + qY~'^ q- + c (p + qY'^ 5* = ^ n'Anp°~"' q'\ 



n 



where A,. = -, V-; — . 



" (c - n) \n\ 



we can easily show that, since p + q = 1, 



— •^■^" «* s ' 



^^= Sa-^ ^ -— pps-xqqs+x 



x=ps (ps - iv) ! {qs + X) ! ^ 

 = spq 



and that \x\ = a / -^ , if s is large *. 



The mean values a"^ and x are the same as those given by 

 von Schweidler for the case when 5 is very large: but we see that 

 their form is quite independent of any assumptions as to the order 

 of magnitude of s, p, q. But the form of \x\ is not so independent, 

 and no further use will be made of this mean. 



* See Bertrand, Calcul des probabilites, chap. iv. 



