46 BELL SYSTEM TECHNICAL JOURNAL 



are functions of N, n, X, and c. This approximation is not so suitable, 

 however, for most telephone sampling problems in which the pro- 

 portion of defectives may be assumed in general to be far smaller 

 than \. 



We shall now proceed to discuss a few points concerning the analysis 

 of Case II in which instead of assuming w{X) constant we assumed 

 it to be of the form 



fX\ (N - X\ 

 Combining this expression for w{X) with the term ( ) I _ ) 



which appears in the basic formula (1), we have 

 X\ {N-X)\ N\ 



c\{X-c)\ {n-c)\{N-X-n+c)\ X\{N-X)\ 



px(^l_p^N-x 



^)-Or(i-.)--(e::).-'(.-.)™ 



Since only the factors in brackets involve the variable of summation 

 X, the remainder of this expression will cancel out in numerator and 

 denominator, leaving us with 



W{X,, X,) 



X=Xi / M _ y,\ 

 X=N-n-\-c / Aj _ ^ \ 



x=c yX - C 



as the resulting form for (1) with this assumption for iv{X). 



It may be noted that the summation in the denominator above is 

 a complete binomial {p + q)^~" and as such equals unity, so 



X=X2 / AT _ ^ \ 



where p is assumed to be the a priori probability of a defective item 

 as determined from reliable information concerning conditions under 

 which the items are prepared. 



As before when Xi = c and X2 = X we have 



We may be willing in certain cases to admit the binomial form for 



