ELEMENTARY SAMPLING THEORY FOR ENGINEERING 41 



A direct interpretation of the expression for W(c, X) gives us the 

 following interesting 



Theorem A: The a posteriori probability that a universe of A^ items 

 contains not more than X defectives when c defectives have resulted 

 from a random sample of n items is equal to the a priori probability 

 of obtaining at least c + 1 defectives in a random sample of « + 1 

 items from a universe of iV + 1 items of which exactly X -\- \ are 

 defective. This theorem assumes that a priori all values of X are 

 equally likely. 



Writing 5 = X + 1 — /, we obtain from (3) 



X+\\(N+\-X + \ 



1 _ w{c, Z) = 1 - ^=" ^ ' /i^+n "" '—^ • (4) 



N - n 



The right-hand side of this equation is exactly what would have been 

 obtained directly from (3) if we had been dealing with a sample of 

 N — n — \ instead of n and had observed X — c defectives instead 

 of c, since the particular symbol chosen for the variable of summation 

 is immaterial. 



This fact, which follows immediately from physical consideration of 

 the equivalent a priori problem of Theorem A , may be stated as 



Theorem B: If we calculate the probability W that a universe of N 

 items contains not more than X defectives when a sample of n has 

 shown exactly c defectives, then \ — W is the probability that a 

 universe of N items contains not more than X defectives when a 

 sample oi N — n — 1 has shown exactly X — c defectives. 



In making extensive calculations, this relation will serve to cut 

 down the amount of computation considerably, as each calculated 

 value of W may be made to do double duty. For a single calculation 

 either (3) or (4) may be used depending on which involves the shorter 

 summation. 



Another interesting relation also appears when we note that 



X + l\/ N-X \ /w+l\/ N-n 

 t )\n + \ - t) \ t )\X + 1 - / 



N + 1\ /N + 



n+\) \X + 



\) 



which may be proved simply by cross multiplication of the combination 

 factors, writing them in terms of factorials. From this we see that 



W{c, X) = \ - ^ ^ \ ,^,^ ^ ~^ ' • (5) 



[n\) 



