Hypergeometric Distribution

How the hypergeometric distribution works

Setup. A finite population of N items contains K "successes" and N − K "failures". You sample n items without replacement — each item drawn is removed, so successive draws are not independent. Let X be the number of successes among the n drawn. Then X follows the hypergeometric distribution with parameters (N, K, n), and

P(X = k) = C(K, k) · C(N − K, n − k) / C(N, n),

where C(a, b) = a!/(b!(a − b)!) is the binomial coefficient. The intuition: out of C(N, n) equally likely n-subsets of the population, the number that include exactly k successes is "choose k of the K successes" times "choose n − k of the N − K failures". Divide by the total.

The support of X is max(0, n − (N − K)) ≤ k ≤ min(n, K). The lower bound says you cannot have fewer successes than the n-sample's surplus over failures; the upper bound says you cannot have more successes than either the sample size or the population's success count.

Mean, variance, and the finite-population correction

The mean of the hypergeometric is

E[X] = n · K / N.

Identical to a binomial with p = K/N. Order and replacement do not affect the marginal probability that any specific draw is a success — each draw, regardless of position, has probability K/N of being a success (this is a symmetry argument, easy to prove by counting orderings).

The variance is

Var(X) = n · (K/N) · (1 − K/N) · (N − n)/(N − 1).

The first three factors are the binomial variance n·p·(1 − p). The last is the finite-population correction (FPC), equal to 1 when n = 1 and 0 when n = N. The hypergeometric is always at most as variable as the corresponding binomial, and strictly less variable when n > 1. Intuition: each draw without replacement removes uncertainty about what is left, tightening the count.

Worked example — quality control

A lot of N = 100 microchips contains K = 8 defective units (mean defect rate 8%). The QA team randomly samples n = 20 chips for destructive testing. What is the probability that X = 0 defective chips are found — i.e. the lot passes inspection?

P(X = 0) = C(8, 0) C(92, 20) / C(100, 20)
         = 1 · C(92, 20) / C(100, 20)
         ≈ 0.1697.

About a 17% chance the lot passes even though 8% are defective. Compare to a binomial approximation (sampling with replacement, p = 0.08, n = 20): P(X = 0) = 0.92²⁰ ≈ 0.1887 — slightly higher because binomial overstates variability. The sampling fraction n/N = 20/100 = 20% > 5%, so hypergeometric matters here.

If instead n/N were small — say N = 10,000 with K = 800 defective and n = 20 sampled — the binomial approximation P(X = 0) ≈ 0.1887 is accurate to 4 decimal places, because removing 20 chips from 10,000 barely affects subsequent probabilities.

Worked example — lottery odds

A 6/49 lottery: N = 49 numbered balls, K = 6 of them are your chosen numbers, and the draw is n = 6 without replacement. The number X of your numbers in the draw is Hypergeometric(49, 6, 6).

P(X = 6) = C(6, 6) C(43, 0) / C(49, 6) = 1 / 13,983,816 ≈ 7.15 × 10⁻⁸.
P(X = 5) = C(6, 5) C(43, 1) / C(49, 6) = 6 · 43 / 13,983,816 ≈ 1.84 × 10⁻⁵.
P(X = 4) = C(6, 4) C(43, 2) / C(49, 6) = 15 · 903 / 13,983,816 ≈ 9.69 × 10⁻⁴.
P(X = 3) = C(6, 3) C(43, 3) / C(49, 6) = 20 · 12341 / 13,983,816 ≈ 0.01765.

Expected number matched is 6 · 6 / 49 ≈ 0.735. About 1 in 13 tickets matches 3, the cheapest payout tier; jackpot 1 in 14 million.

Variants and relatives

• Negative hypergeometric. Sample without replacement until r failures occur; let X count successes drawn. Models "draw until you've rejected r times". Useful when the stopping criterion is failure count rather than sample size.
• Multivariate hypergeometric. Population partitioned into c categories with counts K₁, …, K_c summing to N; draw n without replacement, count successes X₁, …, X_c in each category. PMF involves a product of binomial coefficients over a single big denominator. Used in genetics (allele counts), urn models, multi-class enrichment.
• Fisher's noncentral hypergeometric. Generalises by adding a parameter ω that gives different sampling weights to successes vs failures. Used to model retrospective sampling in case-control studies.
• Wallenius noncentral hypergeometric. Sequential sampling without replacement where the relative odds change at each step. Different from Fisher's noncentral.
• Beta-binomial. The compound distribution if the success probability has a Beta prior — used when the population's K/N itself is uncertain. Approaches hypergeometric in the conjugate-posterior limit.

JavaScript — hypergeometric PMF, mean, and sampling

// Log binomial coefficient for numerical stability with large N
function lnChoose(n, k) {
  if (k < 0 || k > n) return -Infinity;
  let s = 0;
  for (let i = 0; i < k; i++) s += Math.log(n - i) - Math.log(i + 1);
  return s;
}

function hypergeoPMF(N, K, n, k) {
  return Math.exp(lnChoose(K, k) + lnChoose(N - K, n - k) - lnChoose(N, n));
}

function hypergeoMean(N, K, n) { return n * K / N; }

function hypergeoVar(N, K, n) {
  return n * (K / N) * (1 - K / N) * (N - n) / (N - 1);
}

// Sample one draw without replacement
function sampleHypergeo(N, K, n) {
  let successes = K, total = N, x = 0;
  for (let i = 0; i < n; i++) {
    if (Math.random() < successes / total) { x++; successes--; }
    total--;
  }
  return x;
}

// Lottery probabilities
console.log(hypergeoPMF(49, 6, 6, 6));   // ≈ 7.15e-8  (jackpot)
console.log(hypergeoPMF(49, 6, 6, 5));   // ≈ 1.84e-5  (match-5)

// Quality control: 100 chips, 8 defective, sample 20
console.log(hypergeoPMF(100, 8, 20, 0));   // ≈ 0.1697 (P(no defectives))
console.log(hypergeoMean(100, 8, 20));     // 1.6 expected defectives

Why the PMF formula works

Count the favourable outcomes and divide by the total. There are C(N, n) equally likely unordered subsets of size n drawn from the population. To count subsets with exactly k successes: choose which k of the K successes are included (C(K, k) ways), then choose which n − k of the N − K failures are included (C(N − K, n − k) ways). Multiply, divide by C(N, n) — that is the PMF.

An equivalent ordered derivation: each ordering of n draws is equally likely. There are P(N, n) = N!/(N − n)! orderings. Fix a particular pattern of k successes (e.g. SSSFF for k = 3, n − k = 2) — this has K · (K − 1) · (K − 2) · (N − K) · (N − K − 1) orderings. There are C(n, k) such patterns. Divide P(N, n) into ordered count gives the same result. Either viewpoint works; the unordered count is cleaner.

When to reach for hypergeometric

• Sampling without replacement from a fixed population. Quality control by destructive testing, audit sampling, capture-recapture surveys — anywhere the population is finite and items are not returned.
• 2×2 contingency tables with small or fixed margins. Fisher's exact test uses the hypergeometric exact null distribution; the only valid test when expected cell counts are below 5.
• Card games and combinatorial probability. Poker, Bridge, Magic: the Gathering — drawing without replacement is the rule. Hypergeometric gives exact hand probabilities.
• Survey sampling with high sampling fraction. Polling a small town (n/N > 5%), auditing a hospital ward, sampling a small focus group — finite-population correction matters.
• Capture-recapture population estimation. Tag and release, then recapture; the count of tagged animals in the recapture is hypergeometric. Lincoln-Petersen estimator inverts this.
• Gene-set enrichment analysis. Does a gene set overlap a query list more than chance? Test with a hypergeometric tail probability (one-tailed Fisher's exact).

Common pitfalls

• Using binomial when n/N is large. Binomial overestimates variance and undercounts tight cells. Always check the sampling fraction; if n/N > 5%, use hypergeometric.
• Forgetting the finite-population correction. Standard errors for sample proportions in survey sampling include √(FPC). Omit it and you overstate uncertainty by up to 100% when n approaches N.
• Confusing K with the unknown. In quality control, K is the (unknown) number of defectives we want to infer; in lottery, K is known. Make sure you know which parameter is the inferential target.
• Range errors. X must satisfy max(0, n − (N − K)) ≤ X ≤ min(n, K). The lower bound is non-trivial when n is large relative to the failures. A negative lower bound makes no sense.
• Asymptotic approximations without checking conditions. Hypergeometric → binomial only when n/N is small; → Poisson when n is large but np is bounded; → Normal when all of n, K, N − K are large. Use the right limit for the regime.
• Symmetry confusion. Hypergeometric(N, K, n) and Hypergeometric(N, n, K) have the same distribution — drawing n with K successes vs drawing K from a population of N with n successes labelled. Useful for code re-use but a frequent source of mislabelling.

Applications

Fisher's exact test — 2×2 contingency tables

Suppose 100 patients (50 treated, 50 control) experience outcomes. The treated group sees 12 successes, the control 6. Is treatment effective? Under the null hypothesis of independence, conditioning on margins, the count in the treated-success cell is Hypergeometric(100, 18, 50). The p-value is the tail probability P(X ≥ 12) — a finite sum of hypergeometric PMFs. Exact regardless of sample size; the gold standard when chi-squared's approximation is unreliable (expected cell count < 5).

Capture-recapture (Lincoln-Petersen estimator)

Tag K = 100 fish in a lake and release. Later catch n = 150 fish; find X = 20 tagged. Under random mixing, X ~ Hypergeometric(N, 100, 150). The MLE for the unknown lake population is N̂ ≈ n · K / X = 150 · 100 / 20 = 750 fish. Confidence intervals from hypergeometric tails. The Schnabel multi-sample generalisation and Cormack-Jolly-Seber survival models extend this. Used in wildlife biology, epidemiology (estimating disease prevalence from positivity rates), and human population estimation.

Quality control — acceptance sampling

A lot of N items is accepted if a random sample of n contains at most c defectives. Under hypergeometric assumption, the lot's operating characteristic curve — P(accept) as a function of true defect rate K/N — is a sum of hypergeometric PMFs over k = 0 to c. Tables in MIL-STD-105 (now ANSI/ASQ Z1.4) for accept-reject criteria are entirely built on hypergeometric calculations.

Bridge and Poker hand probabilities

Drawing without replacement from a deck. P(four-of-a-kind in 5-card hand) = C(13, 1) · C(4, 4) · C(48, 1) / C(52, 5) = 624 / 2,598,960 ≈ 0.000240 — that's the multivariate hypergeometric with the appropriate cell structure. All canonical poker probabilities reduce to hypergeometric calculations.

Gene-set enrichment analysis (bioinformatics)

From an experiment, you identify n significant genes out of a genome of N. A gene set of interest contains K genes; X of the n significant genes fall in the set. Test enrichment with a one-tailed Fisher's exact P(X ≥ x_obs) — a hypergeometric tail. Tools like DAVID, GSEA-CLI, and enrichr all use this.