Hypergeometric Distribution Calculator

The 10% rule, evaluated rather than remembered

Courses teach that a binomial is an acceptable stand-in when the sample is under about 10% of the population. The rule is sound; what is usually missing is any sense of what it costs when you break it.

The tool’s default draws 20 items from a batch of 50 — a sampling fraction of 40%, well past the rule. The correct probability of exactly 4 successes is 0.2801. A binomial says 0.2182. That is 22% wrong, in a direction that matters: the binomial understates the chance of the outcome near the mean.

Change N to 10,000 with the same proportion and the sampling fraction becomes 0.2%. Now the two agree to 0.06%, which is far below anything you would act on.

That is the whole rule, made concrete. Below about 10% the dependence between draws is too small to matter; above it, it is the dominant feature.

The practical consequence appears in two familiar places. Polling 1,000 people from millions is binomial — the fraction is a rounding error. Inspecting 20 items from a box of 50 is not — and acceptance sampling standards use the hypergeometric for exactly this reason.

Why removing an item changes everything

The mechanism is worth seeing directly, because it explains both the shape of the distribution and the variance correction.

Take a deck of 52 with 4 aces. The chance the first card is an ace is 4/52. If it was, the chance the second is an ace is now 3/51, not 4/52 — you removed one of each. If it was not, the chance is 4/51, slightly better.

So each draw carries information about what is left. The trials are not independent, which is precisely the condition a binomial requires.

The effect is self-correcting, and that is the interesting part. Drawing a success makes the next success less likely; drawing a failure makes it more likely. The composition pulls back toward its average, so results cluster nearer the mean than a binomial would predict.

That is exactly what the variance says. Both distributions have the same mean, n·K/N. The hypergeometric variance is the binomial variance multiplied by (N−n)/(N−1), a factor that is always at most 1.

Two limits check out. Draw the whole population, n = N, and the factor is zero — you have counted every item, so there is no uncertainty left. Let N grow with n fixed and the factor approaches 1, which is the binomial recovering as the population becomes effectively infinite.

The same correction you meet in sample sizing

(N−n)/(N−1) is not a curiosity local to this distribution. It is the finite-population correction, and it is the same factor that reduces a required sample size when the population is small.

The Sample Size Calculator applies its square root to a margin of error, and it is the reason surveying 1,000 people from a closed group of 2,000 gives a tighter result than surveying 1,000 from a country.

Seeing it in both places is worth the moment it takes, because it makes the underlying idea concrete: sampling without replacement gives you more information per observation. Every item you inspect is one you will never have to wonder about again.

Its size follows the same pattern in both settings. At a 1% sampling fraction the factor is 0.99 and nobody cares. At 10% it is about 0.90, so the variance is a tenth lower — the boundary where the rule sits. At 50% it is about 0.50, halving the variance. At 100% it is zero.

One asymmetry worth noting: the correction reduces the variance, but the hypergeometric and binomial probabilities can differ in either direction depending on where k sits. Near the mean the hypergeometric is higher, because outcomes are pulled inward; far out in the tails it is lower.

The values it can and cannot take

The hypergeometric has boundaries a binomial does not, and they are easy to miss.

k cannot exceed K, the number of successes in the population. You cannot draw 6 aces from a deck containing 4, and the probability is exactly zero rather than very small.

k cannot exceed n, obviously.

And there is a lower bound: k ≥ n + K − N. This is the one people miss. If you draw 45 cards from a 52-card deck, you must have at least 45 + 4 − 52 = −3, so zero — fine. But draw 50 cards and you must have at least 2 aces, because only 2 cards remain undrawn. The probability of fewer is exactly zero.

A binomial has no such constraint: it allows any k from 0 to n at every parameter setting, because it imagines an inexhaustible pool. That is the modelling difference stated as a fact about the support.

The tool draws only the achievable range, so the boundaries are visible rather than implied by very small bars at the edges.

Where it is the right model

Anywhere the pool is finite, known, and not replenished.

Acceptance sampling. Inspecting a sample from a delivered batch and deciding whether to accept it. The batch is finite and the sample is often a large share of it, which is exactly the regime where the binomial fails. Sampling standards are built on this distribution.

Card games and lotteries. The canonical case. The chance of exactly two aces in a five-card hand is 0.0399 — C(4,2)·C(48,3)/C(52,5), which the tool computes directly.

Committee and jury composition. Selecting people from a group without repetition. A standard way to assess whether an observed composition is plausibly random.

Capture-recapture. Tag a number of animals, release them, catch a second sample, and count how many are tagged. The count is hypergeometric, and inverting it estimates the population size. The same logic estimates the size of software defect populations.

Gene set enrichment. Asking whether a set of genes contains more members of a pathway than chance would give. This is where Fisher’s exact test comes from — it is the hypergeometric applied to a 2×2 table, which is why that test is exact rather than approximate.

Its relationship to Fisher’s exact test

One connection worth making, because it explains why a widely-used test carries the word “exact”.

A 2×2 contingency table with fixed row and column totals has only one free cell. Under the null hypothesis of no association, the count in that cell follows a hypergeometric distribution.

So Fisher’s exact test is the hypergeometric evaluated over every table at least as extreme as the one observed. There is no approximation anywhere, which is why it stays valid at any cell count, including the small ones where a chi-square test breaks down.

That is the reason the standard advice is to use Fisher when any expected cell count drops below about 5. It is not that chi-square becomes slightly worse — it is that Fisher was never approximating in the first place.

The cost is computational: the number of tables to enumerate grows quickly, which is why chi-square dominated for as long as the arithmetic was done by hand.

Computing it without overflowing

The formula is three binomial coefficients, and computed naively it fails at modest sizes.

C(N, n) for a population of a few hundred already exceeds what a double can represent. Since the formula divides one enormous number by another, evaluating them separately gives infinity over infinity — NaN — while the answer itself is an ordinary probability between 0 and 1.

The fix is the same one the binomial page uses: work in logarithms. Each coefficient becomes a difference of log-gamma values, the three combine by addition and subtraction, and a single exponential at the end brings the result back into range.

Cumulative probabilities are summed term by term over the achievable range, which is short by construction — it can never exceed min(n, K) − max(0, n+K−N) + 1 terms — so there is no need for a closed form here.

The suite asserts that the PMF sums to exactly 1 across the support and that values outside the support return zero rather than a small non-zero artefact, which is the failure mode a log-space implementation is most likely to produce.

Sources and methodology

References for the distribution and its use in sampling.

• NIST/SEMATECH e-Handbook — the hypergeometric distribution
• NIST/SEMATECH e-Handbook — acceptance sampling, where the finite batch makes this the correct model
• R documentation for dhyper — the reference parameterisation this page follows
• Binomial Distribution Calculator — the with-replacement version, and when it is a fair stand-in
• Sample Size Calculator — where the same finite-population correction reduces a required sample
• How we calculate — methodology

Method. The three binomial coefficients are combined in log space through log-gamma and exponentiated once at the end, because C(N, n) overflows a double at populations in the low hundreds and the naive form returns NaN for answers that are ordinary probabilities. Cumulative values sum over the achievable support only, which is bounded by min(n, K) − max(0, n+K−N) + 1 terms. The binomial comparison is computed live on the same inputs rather than described, so the 22%-at-40% and 0.06%-at-0.2% figures quoted above are the tool’s own output. The suite asserts that the PMF sums to 1 across the support, that values outside the support return exactly zero rather than a small log-space artefact, and that the two-aces-in-five-cards case matches the hand calculation C(4,2)·C(48,3)/C(52,5). That engine is verified on every change against 88 assertions. The count and the per-case breakdown are published on the formula verification page.

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