Hypergeometric Distribution Table (Complete PMF & CDF Lookup Chart)

Valid Range of x

x cannot exceed the number of successes available (K), and x cannot be so small that the remaining draws cannot all come from non-successes (N−K).

Expected Value & Variance

The factor (N−n)/(N−1) is the finite population correction, reducing variance versus the binomial.

Step 1 — Identify Your Parameters

Determine N (total population), K (successes in population), n (sample size), and the value of x you are interested in. Match these to a scenario in the table or use the calculator above with your own values.

Step 2 — Find the Row for x

Locate the row where x equals the number of successes you want to evaluate. The PMF column gives the exact probability of that specific outcome. The CDF column gives the probability of getting that many or fewer successes.

Step 3 — Use the CDF for Inequality Probabilities

For probabilities of the form P(X ≤ x), read the CDF column directly. For P(X ≥ x), calculate 1 − P(X ≤ x−1). For P(a ≤ X ≤ b), calculate P(X ≤ b) − P(X ≤ a−1).

Step 4 — Verify the Range of x

The hypergeometric distribution has a restricted support. If you ask for a value of x below max(0, n+K−N) or above min(n, K), the probability is exactly 0. The table only shows rows for x values with non-zero probability.

Microsoft Excel / Google Sheets

Note: Excel argument order is (x, n, K, N) — not the same as the mathematical formula. The 5th argument TRUE gives the CDF; FALSE gives the PMF. Google Sheets uses the same HYPGEOM.DIST syntax.

R

R uses m (successes in population), n (non-successes = N−K), k (sample size). Be careful — R's "n" is not the same as the mathematical formula's n (which is the sample size).

Python (SciPy)

SciPy's hypergeom uses M=N (population), n=K (successes in population), N=sample_size. The argument naming differs from standard notation — always verify by checking M > n and N ≤ M.

Quality Control & Manufacturing

Lot acceptance sampling: inspecting a batch of n items from a lot of N to determine whether to accept or reject the lot, knowing that K items are defective. Used widely in ISO standards and acceptance sampling plans (MIL-STD-1916).

Clinical Trials & Healthcare

Randomizing patients into treatment arms without replacement, evaluating genetic marker prevalence in selected subgroups, or assessing adverse event rates in subpopulations. Fisher's exact test for 2×2 contingency tables is directly derived from the hypergeometric distribution.

Ecology & Biological Research

Mark-recapture experiments estimate population size. A researcher marks K animals, releases them, then recaptures n. The number of marked animals in the recapture follows the hypergeometric distribution. This is also the basis for the Lincoln-Petersen population estimator.

Survey Sampling & Social Science

When sampling from small, well-defined populations (a classroom, a department, a committee), the hypergeometric provides exact probabilities. Applies to stratified sampling without replacement, oversampling of minority groups, and committee composition analyses.

Card Games & Combinatorics

Computing exact probabilities in poker (5-card hands from a 52-card deck), bridge, and Magic: The Gathering deck analysis. The hypergeometric is the standard model for calculating hand probabilities in any card game where cards are dealt without replacement.

Auditing & Fraud Detection

Auditors sample financial records from a finite set of transactions. The hypergeometric distribution tells auditors the probability of detecting a given number of errors when sampling without replacement, enabling statistically sound sampling plans under auditing standards such as PCAOB AS 2315.

Mistake 1 — Using Binomial When Sampling Is Without Replacement

The most frequent error: applying the binomial formula to a without-replacement scenario. If you draw cards from a deck, inspect items from a finite lot, or sample people from a small group, successive draws are not independent. The hypergeometric is required. Only use binomial when n/N ≤ 0.05 as an approximation.

Mistake 2 — Confusing PMF and CDF

P(X = 2) and P(X ≤ 2) are completely different quantities. The PMF gives the probability of exactly that value; the CDF gives the probability of that value or any smaller value. "At most 2" → CDF. "Exactly 2" → PMF. "At least 2" → 1 − CDF at x=1. Mixing these up leads to answers that are off by a factor of 2–5.

Mistake 3 — Wrong Argument Order in Software

Excel, R, and Python all use different argument orderings for their hypergeometric functions. In Excel: (x, n, K, N). In R: (x, K, N−K, n). In SciPy: (M=N, n=K, N=sample_size). Entering arguments in the wrong order produces silent wrong answers. Always verify with a simple case where you know the result.

Mistake 4 — Requesting Probabilities Outside the Valid Range

If you ask for P(X = 6) when K = 4 (you can't draw more successes than exist in the population), the probability is 0 — but some calculators return errors rather than zero. Always verify x is within max(0, n+K−N) ≤ x ≤ min(n, K). Requests outside this range are physically impossible.

Mistake 5 — Confusing N (population) with n (sample)

The notation N (population size) and n (sample size) are case-sensitive and represent completely different quantities. N is always larger than n. K is always ≤ N. x is always ≤ min(n, K). Swapping N and n in calculations produces completely wrong probabilities and is one of the most common exam errors.

Is the hypergeometric distribution discrete or continuous?

The hypergeometric distribution is discrete. X can only take integer values within its valid range. It belongs to the family of discrete probability distributions alongside the binomial, Poisson, and geometric distributions. This is why it has a PMF (not a PDF, which applies to continuous distributions).

What is Fisher's exact test and how does it relate to the hypergeometric?

Fisher's exact test, developed by Ronald Fisher, uses the hypergeometric distribution to test for independence in a 2×2 contingency table. Given fixed row and column totals, the probability of the observed table (and all more extreme tables) is computed using the hypergeometric PMF. It is the preferred test over chi-square when sample sizes are small or expected cell counts are below 5. See Hypothesis Testing for more on Fisher's exact test.

What is the finite population correction factor?

The finite population correction (FPC) is the factor (N−n)/(N−1) that appears in the hypergeometric variance formula. It quantifies how much the variance is reduced compared to binomial sampling because once an item is drawn, it cannot appear again. As n → N (sampling nearly the whole population), FPC → 0, meaning there is almost no variance — you would know almost exactly how many successes there are. As n/N → 0, FPC → 1, and the hypergeometric variance approaches the binomial variance.

Can the hypergeometric distribution be approximated by the normal distribution?

Yes, for large sample sizes the hypergeometric distribution is approximately normal with mean μ = n(K/N) and standard deviation σ = √[n(K/N)(1−K/N)(N−n)/(N−1)]. This approximation works well when both n(K/N) ≥ 5 and n(1−K/N) ≥ 5. For smaller samples, use the exact hypergeometric table or calculator. The normal distribution approximation can be improved with a continuity correction of ±0.5.

What is the multivariate hypergeometric distribution?

When the population has more than two categories (not just successes and failures), the multivariate hypergeometric distribution generalizes the standard formula. If the population has categories of sizes K₁, K₂, …, Kc (summing to N), the probability of drawing exactly x₁ from category 1, x₂ from category 2, etc., in a sample of n is: P = [C(K₁,x₁) × C(K₂,x₂) × … × C(Kc,xc)] / C(N,n). This arises in race/gender composition analyses, multi-arm clinical trials, and multi-species ecological sampling.

Is the hypergeometric distribution used in genetics?

Yes, extensively. Gene set enrichment analysis (GSEA) uses the hypergeometric distribution to test whether a list of differentially expressed genes is enriched for a particular biological pathway. Given N total genes, K genes in a pathway, a hit list of n genes, and x genes in the intersection, P(X ≥ x) from the hypergeometric distribution gives the p-value for enrichment. Tools like GOstats, clusterProfiler, and GOrilla use this exact calculation. This is one of the most widespread applications of the hypergeometric distribution in modern biology.

Binomial Distribution

Models the number of successes in n independent Bernoulli trials with constant probability p. The hypergeometric converges to the binomial as N → ∞ with K/N → p. The binomial distribution is the sampling-with-replacement counterpart.

Negative Hypergeometric Distribution

Describes the number of non-successes encountered before achieving the r-th success in without-replacement sampling. Analogous to the relationship between the binomial and negative binomial distributions. Less commonly tabulated but used in sequential sampling inspection.

Poisson Distribution

When N is large, K is small, n is small, and K/N is small, the hypergeometric can be approximated by the Poisson with λ = nK/N. This is the approximation commonly used in rare-event problems like defect rates in high-volume manufacturing. See our Poisson distribution guide.