38 BELL SYSTEM TECHNICAL JOURNAL 



Case II 

 Another assumption is suggested by the following considerations 

 which enter into many of the standard works on probability theory. 

 Assume that the lot or universe in question was itself drawn at random 

 from an extremely large stock or major universe in which the pro- 

 portion of defective items was p. Under these conditions the a 

 priori probability w{X) that our universe of N items would contain 

 exactly A'' defective items would be given by the expression 



w{x) = (^)/'^(i - pr- 



Using this expression for w{X) in the fundamental formula (1), 

 we obtain, by a process given in detail under the heading Appendix, 

 the formula 



WiXu X,) = 2 ( X - c ) ^""'^^^ ~ P)"'""''^'^ 



X — Xi \ "* / 



which for AT 1 = c and A2 = A" reduces to 



X-e / AT _ ^ \ 



w{c,x) = Z(^ )p'(i - pr"'-\ 



which is precisely the expression for the a priori probability that the 

 remaining N — n items which we did not inspect contain not more 

 than the X — c defectives which together with the c we have observed 

 would assure us of a satisfactory universe. 



In this form W{c, X) turns out to be a simple binomial which, 

 when A^ — w is large and p is small, may be reasonably approximated 

 by the Poisson Exponential Binomial Limit for which extensive curves 

 and tables already exist ^ and will, therefore, not be included in this 

 article. 



In order to make practical use of the results of this assumption, 

 we must have some knowledge of the appropriate value of the factor 

 p in any given case. This factor should measure the probability 

 that an item, selected at random, will, on inspection, prove to be 

 defective. If a large number of tests have been made in the past 

 on similar items, prepared by essentially the same process, the ratio 

 of the total defectives observed to total items inspected in such tests 

 may be a reasonable figure to use for p. In the case of many manu- 

 factured articles such a ratio ought not to be very difficult to obtain. 

 In certain cases it might be necessary to allow for such factors as 



"" See article, "Probability Curves Showing Poissoii's Exponential Summation," 

 by George A. Campbell, Bell System Technical Journal, January, 1923. 



