Hypergeometric Distribution
The hypergeometric distribution models drawing successes from a finite population without replacement.
Learn the formula with examples.
The Formula
The hypergeometric distribution calculates the probability of getting exactly k successes in n draws from a finite population of size N that contains exactly K successes, without replacement. Unlike the binomial distribution, the hypergeometric distribution does not assume independence between draws.
When you draw without replacement, each draw changes the composition of the remaining population. This makes the hypergeometric distribution the correct model for many real-world scenarios, such as quality control sampling, card games, and lottery calculations.
The mean of the hypergeometric distribution is μ = nK/N. The variance is σ² = n × (K/N) × (1 - K/N) × (N-n)/(N-1). When the population is very large relative to the sample, the hypergeometric distribution approaches the binomial distribution.
Variables
| Symbol | Meaning |
|---|---|
| N | Total population size |
| K | Number of success states in the population |
| n | Number of draws (sample size) |
| k | Number of observed successes in the sample |
| C(a, b) | Binomial coefficient = a! / (b!(a-b)!) |
Example 1
A deck of 52 cards has 4 aces. You draw 5 cards. What is the probability of getting exactly 2 aces?
N = 52, K = 4 (aces), n = 5 (drawn), k = 2 (desired aces)
C(4, 2) = 6, C(48, 3) = 17,296, C(52, 5) = 2,598,960
P(X = 2) = 6 × 17,296 / 2,598,960 = 103,776 / 2,598,960
P(X = 2) ≈ 0.0399 or about 4.0%
Example 2
A box has 20 items, 5 of which are defective. A quality inspector randomly selects 4 items. What is the probability that exactly 1 is defective?
N = 20, K = 5 (defective), n = 4 (inspected), k = 1
C(5, 1) = 5, C(15, 3) = 455, C(20, 4) = 4,845
P(X = 1) = 5 × 455 / 4,845 = 2,275 / 4,845
P(X = 1) ≈ 0.4696 or about 47.0%
When to Use It
Use the hypergeometric distribution whenever you sample without replacement from a finite population.
- Card game probability calculations
- Quality control — testing a sample from a production batch
- Lottery and raffle probability
- Ecology — capture-recapture population estimates