How this calculator works
The numerator counts ways to select the requested successes and the remaining failures. The denominator counts all samples of that size. The finite population factor reduces variance relative to independent sampling. The numerical limits are one million population items and 10,000 draws. Sampling the whole population makes the count certain.
Formula
P(X=k) = C(K,k)C(N-K,n-k)/C(N,n); E[X] = nK/N; Var[X] = n(K/N)(1-K/N)(N-n)/(N-1).
Worked example
With N = 10, K = 4 and n = 3, exactly two successes have probability 36/120 = 0.3. At most one has probability 80/120 = 2/3.
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Frequently asked questions
When should I use binomial instead?
Use binomial for independent sampling or sampling with replacement.
Can all draws be successes?
Only if the population contains at least as many successes as draws.
Why is the variance smaller?
Removing an item changes the remaining composition, creating negative dependence.