What is the hypergeometric distribution?

The hypergeometric distribution models the probability of getting exactly k successes in n draws without replacement from a finite population of size N that contains K successes.

When do I use it instead of the binomial?

Use the hypergeometric when sampling without replacement from a finite population. Use the binomial for independent trials with replacement or very large populations.

What inputs do I need?

You need N (population size), K (number of successes in the population), n (sample size), and k (number of successes observed).

Hypergeometric Distribution Calculator

Model “successes without replacement” — for example, drawing red balls from an urn. Enter N, K, n, k and get the probability and distribution table.

1. Input parameters

Population size (N)

Total items in the population

Number of success states (K)

How many of the N are “success”

Sample size (n)

Drawn without replacement

Observed successes (k)

Exact number you want the probability for

2. Results

P(X = k)

—

Cumulative

P(X ≤ k): —

P(X ≥ k): —

Mean

—

Variance

—

3. Distribution table

All valid k from 0 to n (subject to K and N). Handy for homework / QA.

k	P(X = k)	P(X ≤ k)

Hypergeometric distribution: formula

For a population of N items containing K successes, when you draw n items without replacement, the probability of observing exactly k successes is:


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

where C(a, b) is the number of combinations “a choose b”.

Mean and variance


                E[X] = n · (K / N) 

                Var(X) = n · (K / N) · (N - K)/N · (N - n)/(N - 1)

Where it’s used

Quality control (defectives in a batch)
Card games (drawing particular suits)
Sampling without replacement in surveys

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

\[','\\]

','\

Formula (extracted text)

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

Formula (extracted text)

E[X] = n · (K / N) Var(X) = n · (K / N) · (N - K)/N · (N - n)/(N - 1)

Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
Profile · LinkedIn