Tools/Hypergeometric Calculator

Hypergeometric Distribution Calculator

Drawing without replacement. Use presets for lottery draws, blackjack card counting, and quality sampling. Includes P(X=k), cumulative probabilities, and the PMF chart.

Setup

Preset Population N Successes in population K Sample size n (draws) k (successes in sample)

Result

PMF (probability mass function)

Table

k	P(X=k)	P(X≤k)	P(X≥k)

When to use hypergeometric (vs. binomial)

If you draw cards from a deck, balls from an urn, or items from a batch without replacement, you need the hypergeometric. Binomial assumes replacement, which overstates variance for finite populations. The difference is small for large N, but for lotteries with N ≈ 50, hypergeometric is exact.

Who uses it

Quality engineers doing acceptance sampling (MIL-STD-105 / ANSI Z1.4).
Poker / blackjack players computing the probability of completing a draw.
Lottery enthusiasts checking the math of various ticket strategies (none of which improve odds, for the record).
Ecologists doing capture-recapture population estimates.

Limitations

We cap computations at N ≤ 1000 for browser performance. Larger populations are well-approximated by binomial.
Assumes simple random sampling without replacement. Stratified sampling requires different formulas.