SAMPLING INSPECTION TABLES 19 



sample from the general output of product — a source of supply — and probabilities 

 are correspondingly based on (b). 



Finite Universe 



The probability of finding m defects in a random sample of n units drawn from 

 a finite universe (lot) of N pieces in which the number of defects is M = pN , 

 is given exactly by 



rm.n.N.M - ffCn-m <-m • Uj 



When p < 0.10, a good approximation to (1) is given by the m + 1st term of 



Kn\ nl" 

 1 — - J + - , 



Pm,n.^.M = P^.^.M = C ^ " ^ V " Cf, ) • (l') 



n 

 When p < 0.10 and when — < 0.10, a good approximation to (1) is given by 



the m + 1st term of the Poisson exponential distribution, 



Pm,n,N,M = Pm.vn ^^ ', ' • (1 ) 



ml 



These are general equations apphcable for any fraction defective, p, but are 

 used in this paper only for the specific case where p = pt, the lot tolerance fraction 

 defective, and where in turn M = ptN. 



The Consumer's Risk Pc, is the probability of meeting the acceptance criteria — 

 c, for single sampling, and Ci and C2, for double sampling — in samples drawn from 

 a lot of N pieces containing exactly the tolerance number of defects M = ptN. 



For single sampling, 



rn=c 



Pc = X) Pr,i,n.N.M (whcn p = pt) . (2) 



m=0 



For double samphng, 



m=c i Tn=C2— cx— 1 



Pc = 2^ -Pjn.nj.A', Af + Pci+l,ni,N,M 2_j Pm,n2<N—ni,M—ci—l 



m=0 m=0 



m=C2— ci— 2 

 ■\- Pci+2,ni,S,M 2_^ Pm,ni,N—ni,M—ci—2-\- •" 



m=0 



+ Pc2,ni,N,M Po,n2,N-ni,M—C2 (whenp = pt). (3) 



Values of Pc in equations (2) and (3) are given approximately by substi- ] (2') 

 tuting Pm,'l,M or Pm.pn for Pm,n,N,M throughout, in accordance with equa- I (2") 



tions (1') and (1"), using p = pi. The resulting equations will be referred (3') 

 to as (2'), (2"), (3') and (3"), respectively. J (3") 



