20 BELL SYSTEM TECHNICAL JOURNAL 



Infinite Universe 



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

 from an infinite universe (general output of uniform product) in which the frac- 

 tion defective is /», is given exactly by the m + 1st term of the expansion of the 

 binomial, [(1 - p) + pY, 



Pm.n.p = a(l - py-^p"". (4) 



When p < 0.10, a good approximation to (4) is given bj' the w + 1st term 

 of the Poisson exponential distribution, 



_ _ e-vr.^pnr .4. 



J^ m.n,p — J^ m,pn — , • - V.^ / 



The probability of meeting the acceptance criteria — c, for single sampling, 

 and ci and ("2 for double sampling — in samples drawn from submitted product 

 having a fraction defective of p, is termed the probability of acceptance. Pa. 

 For single sampling, 



■t^a ^^ / ^ I^m,n,p • \^) 



For double sampling, 



m=ci m=C2— Ci— 1 m=C2—ci—2 



Pa = / ^ Pm,ni,p "T -t ci+l,n j.p / ^ Pm,n2,P I Pci+2,ni,p / ^ "m,n2,p 



j»=0 »n=0 m=Q 



+ ••• + -r C2ini,p -1 0,n2.P • (.O) 



Values of Pa in equations (5) and (6) are given approximately by substituting 1 

 Poisson exponential probabilities, Pm,pn , for Pm,n,p throughout in accord- ( (5') 

 ance with equation (4'). The resulting equations will be referred to as | (6') 

 equations (5') and (6'), respectively. J 



The Poisson exponential approximation is used in subsequent paragraphs 

 wherever probabilities in sampHng from an infinite universe apply. Tables^ 

 and charts^' '" are available from which these probability values (single term 

 values, or cumulative values for "c or less defects") may be read directly.* Fig- 

 ure 6 gives a cumulative probability chart for the Poisson exponential distribution, 

 which is widely useful in the solutions involved. 



The Producer's Risk, Pp, is the probability of failing to meet the acceptance 

 criteria in samples drawn from product of process average (p) quality. Using 

 p = p in equations (5) and (6), 



Pp = I - Pa (when p = p). (7) 



Lot Quality Protection 

 Single Sampling 



Given: Lot Size (N), lot tolerance fraction defective (pt), Consumer's Risk 

 {Pc = 0.10), process average fraction defective (p). 



* In this work use was made of more complete tables, giving cumulative probabilities 

 for pn values up to 100, prepared by Office of the Switching Theorj' Engineer, Bell 

 Telephone Laboratories. 



