Hypergeometric — quality control

Sample testing of a batch of items is a classic industrial problem modelled with the hypergeometric: a sample is drawn without replacement and the number of defective items is counted.

Example — Quality control

A batch of $1000$ items contains $20$ defective ones (a number unknown to the inspector). A sample of $50$ items is inspected without replacement. What is the probability of finding $0$ defective items?

With $N=1000$, $K=20$, $n=50$, $k=0$: $P(X=0) = \frac{\binom{20}{0}\binom{980}{50}}{\binom{1000}{50}} = \frac{\binom{980}{50}}{\binom{1000}{50}}\approx 0{,}358.$

There is therefore about a $36\%$ probability that the sample turns out “clean” even though there are defective items in the batch: an outcome that helps to size testing plans (how many items to inspect to have a given guarantee).

Links

Topics: Probability distributions
Concepts: Hypergeometric distribution
Methods: Hypergeometric distribution
Skills: Probability calculation · Modelling