Hypergeometric Distribution

Count marked items in finite samples without replacement, explore exact event probabilities, and compare the variance with a replacement model.

Hypergeometric Distribution

Concept Overview

Take a fixed-size random sample from a finite collection containing marked and unmarked items. If items are not returned, draws depend on one another. The hypergeometric distribution describes the number of marked items selected. Here “marked” names a category, not a desirable outcome: it could mean defective parts, tagged animals or red cards.

Mathematical Definition

Let N be the population size, K the number marked and n the sample size. Every subset of n distinct items has the same chance of selection. X counts marked items in the subset; C(a, b) denotes a binomial coefficient.

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

max(0, n − (N − K)) ≤ k ≤ min(n, K)

F(x) = P(X ≤ x) = Σ_{k ≤ x} P(X = k)

P(X ≥ x) = Σ_{k ≥ x} P(X = k)

Choose k of the K marked items, then n − k of the N − K unmarked items. Their product counts favorable subsets. Divide by all C(N, n) subsets. Integer counts outside the feasible interval have probability zero. The upper-tail event includes equality, so P(X ≥ x) equals 1 − F(x − 1), rather than 1 − F(x).

Key Concepts

Same mean, different dependence

Each draw has marginal marked probability p = K/N. After observing a marked item, the next conditional probability becomes (K − 1)/(N − 1). Let I_i indicate whether draw i is marked. For N > 1, distinct indicators have covariance −p(1 − p)/(N − 1). Adding their means and covariances gives:

X = Σ_i=1ⁿ I_i, E[X] = np

Var(X) = np(1 − p)(N − n)/(N − 1)

With replacement: Y ~ Binomial(n, p)

E[Y] = np, Var(Y) = np(1 − p)

For N > 1 and 1 ≤ n ≤ N the finite population correction (N − n)/(N − 1) lies between zero and one. A census (n = N) removes uncertainty. With fixed n and K/N approaching p as N grows, the hypergeometric distribution approaches the binomial distribution. The module overlays this replacement model; it does not treat it as an exact substitute.

A counted example

For N = 10, K = 4 and n = 3 there are C(10, 3) = 120 subsets. Exactly one marked item occurs in 4 × C(6, 2) = 60 subsets, so its probability is 1/2. Zero marked items occurs in C(6, 3) = 20 subsets. Thus P(X ≤ 1) = 80/120 = 2/3 and P(X ≥ 1) = 100/120 = 5/6. The mean is 1.2 and the variance is 3(0.4)(0.6)(7/9) = 0.56.

Boundary cases and scope

An empty sample has X = 0. If K = 0 or K = N, every sample count is fixed. A one-item population also has variance zero; the displayed formula must not be evaluated as 0/0. The explorer supports integer N from 1 to 100, K and n from 0 to N, and query x from −100 to 100. Blank, fractional and incompatible inputs are rejected. Numerical probabilities are displayed to six decimal places; a very small positive probability may round to zero. Selection must be uniform without replacement; biased selection requires a different model.

Historical Context

The name reflects a connection with hypergeometric series: the ratio of neighboring nonzero probability masses is a rational function of the count. The finite-sampling interpretation offers a direct way to understand the distribution through counting.

Real-world Applications

Batch inspection: Calculate the chance that a random sample contains a specified number of defective items when the batch composition is known.
Card hands: Count cards of a chosen suit in a hand dealt from a fixed deck.
Tagged populations: Model how many previously tagged animals appear in a uniformly selected recapture sample, under the required sampling assumptions.

Related Concepts

Conditional Probability Tree — changing conditional probabilities explain dependence between draws.
Negative Binomial Distribution — independent trials continue until a target count, rather than a fixed sample without replacement.
Bootstrapping — resampling with replacement permits repeated selection of the same item.

References

Kyle Siegrist, Random: The Hypergeometric Distribution — derivation, moments and replacement comparison.
SciPy documentation: hypergeom — PMF and feasible support. Its parameter letters differ from this explorer.

Hypergeometric Distribution