# Sampling Plans (ISO 2859-1)(BS6001-1) - Probability of defects within a lot

#### Stuart Andrews

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How do I work out the probability of defects within a lot?
For example, if I'm using percent nonconforming to Inspection Level II, and an AQL of 1% with a double sampling plan. My lot size is 100 pieces and I've found no defects from my 20 samples.

R

#### Rick Goodson

Stuart,

The calculation for probability of defects within a lot of 100 pieces, a sample of 20 and 0 defects found is based on the Hyoergeometric Probability Distribution, since the lot is finite and the random sample is taken without replacement. The formula for finding the probability 'P' of 'd' defects in a sample of size 'n' from a lot of size 'N' is:

P(d) = 'Combination of all defective units' multiplied by the 'Combination of all good units' divided by the 'Combination of all units'

An easier method of finding the probabilities would be to use the Operating Characteristic curve (OC curve) for the sampling plan you are using. In the sampling plan find the 'Code Letter' for your universe of 100 and an Inspection II. Find the OC curve for that code letter. The graph will have curves for various AQL's. The 'x' axis is the percent defective of the lot and the 'y' axis is the probabilit of acceptance. Using the intersect of horizontal and vertical lines to the curve you can determine the probability of acceptance of a lot based on various percent defectives. In essence, for each percent defective you can look at the probability of accepting the lot based on your smapling result.

A problem inherent with Operating Characteristic curves is that most are based on an assumption of an 'infinite' lot rather than the 'finite' lot size as you have. The curves therefore are not based on hypergeometric probabilities although the Poisson or binomial probability distributions often give acceptable results.

There is an excellent discusion on sampling plans in 'Statistical Quality Control' by Grant and Leavenworth, ISBN 0-07-114248-7.

There are also some good referemces on the web:
https://www.itl.nist.gov/div898/handbook/pmc/section2/pmc24.htm has a good description of how a double sampling plan is created and therefore some insight into the problems you may run in to.

#### Stuart Andrews

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Rick Goodson, Thank you very much!

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#### lmay0001

HI

i have a problem

SAmple 55, accepted 2 or less, reject 3 or more defect. Hence draw an OC curve for this plan.

what am i supposed to do?? what are the percentage of defects for the binomial?????

Hi

i have a sampling plan 55 2/3 ( accept 2 and reject 3) draw a OC curve for the plan.

how do u draw such a OC curve i dont even know the percent defects how do i draw it?

how can i calculate a binomial with out percent defect ( example 0.5%)

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