80 BELL SYSTEM TECHNICAL JOURNAL 



probability that the event will happen every time is pppp p, 



where the factor p appears n times; that is the probability is p". 

 The probability that the event will occur (« — 1) times in succession 

 and then fail is p""'^ q. 



But if the order of occurrence is disregarded, this last combination 

 may arrive in n different ways; so that the probability that the event 

 will occur (« — 1) times and fail once is n^"~^ q. Similarly, the prob- 

 ability that the event will happen {n — 2) times and fail twice is 

 ^"~2 g2 multiplied by n{n — l)/2, etc. That is, the probabilities 

 of the several possible occurrences are given by the corresponding terms 

 of the binominal expansion of {p + qV- Let 



p = p--\- (i)^""^? + Qp"-v + 



+ C + l)^^^V-^-^ + (^)r2"-S (1) 



where ( J means n {n — 1) (w — 2) {n — x-\- 1)/(1) (2) (x) . 



Then P = probability that the event happens exactly n times, plus the 

 probability that it happens exactly {n — 1) times . . . plus 



the 

 probability that it happen exactly c times: in other words, 



the 

 probability that the event happens at least c times in n 



trials. 



If the series for P contains few terms it may be computed easily. 

 In general, however, it is impracticable to compute P by means of 

 the above binomial expansion. Other forms for the value of P 

 must, therefore, be developed. 



One of the most convenient approximations for P when p is small 

 has been developed by Poisson. It is known as Poisson's Exponen- 

 tial Binomial Limit and gives the value of P by the following ex- 

 pansion 



P = e-^a'licy. -\- e-^a'+^/ic -f 1)! 



4- e-<'a^+2/(c -F 2) ! ad inf. (2) 



where e = base of natural logarithms = 2.718 a = (np) and 



{c)\ = c(c-l) (c - 2) (c - 3) (3) (2) (1). 



The following Table gives corresponding values of P, a, c satisfying 

 equation (2). 



