Question 1

When should I use the hypergeometric distribution instead of binomial?

Accepted Answer

Use hypergeometric when sampling without replacement from a finite population where K/N is not negligible. Use binomial when: (1) sampling with replacement, or (2) sampling without replacement but the population is much larger than the sample (n/N < 0.05), making the "without replacement" effect negligible. Rule of thumb: if n/N > 5%, use hypergeometric; if n/N ≤ 5%, binomial is a good approximation.

Question 2

What are real-world examples of the hypergeometric distribution?

Accepted Answer

Common applications: (1) Quality control — a shipment of N items contains K defective; you sample n items to test how many defectives k appear. (2) Card games — in a 52-card deck (N=52, K=4 aces), deal n=5 cards; what is the probability of exactly k aces? (3) Clinical trials — if N patients have K who respond to treatment, and n are randomly selected for analysis. (4) Ecology — capture-recapture for estimating population sizes.

Question 3

What is the mean and variance of the hypergeometric distribution?

Accepted Answer

Mean = n × K / N (same as binomial with p = K/N). Variance = n × (K/N) × (1 − K/N) × (N − n)/(N − 1). The factor (N − n)/(N − 1) is the finite population correction factor, which is always ≤ 1. This means the hypergeometric distribution has less variance than the equivalent binomial, reflecting the reduced randomness when sampling without replacement from a finite population.

Question 4

How do I calculate the CDF of the hypergeometric distribution?

Accepted Answer

The CDF P(X ≤ k) is the sum of PMF values: Σ P(X = j) for j = 0 to k. There is no simple closed form, but most calculators and software compute it directly. This calculator computes the exact CDF by summing the PMF terms. For large N, the binomial CDF is a good approximation.

Question 5

What are the constraints on parameters N, K, n, k?

Accepted Answer

Required: N ≥ 1 (total population), 0 ≤ K ≤ N (successes in population), 0 ≤ n ≤ N (sample size). The possible values of k range from max(0, n − (N − K)) to min(n, K). Values outside this range have P(X = k) = 0. For example: if N = 10, K = 3, n = 8, then k must be at least n − (N − K) = 8 − 7 = 1 (you cannot avoid having at least 1 success in a sample of 8 from a population of 10 with 3 successes).

Question 6

Hypergeometric vs Binomial vs Poisson — which to use?

Accepted Answer

Hypergeometric: fixed N, K, n; sampling without replacement; finite population. Binomial: fixed n trials, constant probability p per trial; with replacement (or infinite population). Poisson: events in time/space with rate λ; rare events in large populations. When n/N ≤ 0.05, hypergeometric ≈ Binomial(n, K/N). When n is large and p = K/N is small, Binomial(n, p) ≈ Poisson(np).

Feature	Hypergeometric	Binomial
Sampling	Without replacement	With replacement (or infinite population)
Trial independence	Trials are dependent	Trials are independent
Parameters	N, K, n	n, p
Mean	nK/N	np
Variance	nKM(N−n) / [N²(N−1)]	np(1−p)
Finite pop. correction	Yes: (N−n)/(N−1)	No (= 1)
Use when	n/N > 5%	n/N ≤ 5% (or replacement)

Hypergeometric Distribution Calculator

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Hypergeometric vs Binomial — Key Differences

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