Binomial, Poisson and hypergeometric calculator

Binomial: a fixed number of independent trials, each pass or fail, with a constant probability, such as defectives in a sample from a large lot. Hypergeometric: the same, but sampling without replacement from a small lot, so the probability changes as you draw. Poisson: counts of events in a fixed interval or area, such as defects per unit or failures per month.

When n is large and p is small (a common rule is n ≥ 20 and p ≤ 0.05), the binomial is close to a Poisson with λ = np. Exam questions often expect this shortcut; this calculator gives you both so you can see the difference.

When the sample is a small share of the lot (under about 10%), drawing without replacement barely changes the probability, so the binomial is a good approximation.

For a single sampling plan with sample size n and acceptance number c, P(accept) = P(X ≤ c). Use the hypergeometric for an isolated small lot, binomial for a large lot or a continuing process, and Poisson as the usual approximation.

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