42 BELL SYSTEM TECHNICAL JOURNAL 



If we compare the right-hand sides of (3) and (5), we see that n 

 and X have simply been interchanged, which proves the rather 

 interesting 



Theorem C: The probability W{c, X) that a universe of A^ items 

 does not contain more than X defectives when a sample of n has 

 shown exactly c defectives is equal to the probability W{c, n) that a 

 universe of TV items does not contain more than n defectives when a 

 sample of X has shown exactly c defectives. 



Thus equations (3), (4) and (5) taken together show that we may 

 make three different interpretations of a single calculation. 



Up to this point, all of the analysis has been exact on the basis of 

 the fundamental assumptions. We may now proceed with advantage 

 to consider some approximate relationships which have been for 

 some years of service in the calculation of practical curves and tables 

 for cases where the values of N, n, or X were too great to be handled 

 conveniently by means of the exact formulae. 



Now consider in formula (3) a single term, tti say, where 



rt 



(A^ - n) ! 



X + r\ \ (w + 1 - t)\ /\{N - n - X - 1 + ! 

 ^ + 1 \/ « \7 . n 



{N - X)\ 



where the form of the function F is to be determined. 



To facilitate the consideration of this function we may split it up 

 into three similar parts as follows: 



^^^' ^' ^' '^ ^ ^(A^+1,A'; ' 



where 



