Hypergeometric Distribution — Definition, Formula & Examples

The hypergeometric distribution gives the probability of drawing exactly

k

successes from a finite population when you sample without replacement. It applies whenever you pull items from a group that contains two types (success/failure) and do not put them back.

A discrete random variable $X$ follows a hypergeometric distribution with parameters $N$ (population size), $K$ (number of success states in the population), and $n$ (number of draws) if its probability mass function is $P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$ for $\max(0,\, n+K-N) \le k \le \min(n,\,K)$ .

Key Formula

P(X = k) = \frac{\dbinom{K}{k}\dbinom{N - K}{n - k}}{\dbinom{N}{n}}

Where:

$N$ = 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

How It Works

You use the hypergeometric distribution when three conditions hold: the population is finite, each member is classified as success or failure, and sampling is done without replacement. To find

P(X=k)

, count the ways to choose

k

successes from

K

, multiply by the ways to choose

n-k

failures from

N-K

, then divide by the total ways to choose

n

items from

N

. The expected value is

E(X) = \frac{nK}{N}

, and the variance is

\text{Var}(X) = n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}

. As

N

grows large relative to

n

, the hypergeometric distribution approaches the binomial distribution.

Worked Example

Problem: A deck contains 20 cards: 6 red and 14 black. You draw 5 cards without replacement. What is the probability of getting exactly 2 red cards?

Identify parameters: Population

N = 20

, successes

K = 6

, draws

n = 5

, desired successes

k = 2

Count favorable outcomes: Choose 2 red from 6 and 3 black from 14.

\binom{6}{2}\binom{14}{3} = 15 \times 364 = 5{,}460

Count total outcomes: Choose any 5 from 20.

\binom{20}{5} = 15{,}504

Compute probability: Divide favorable by total.

P(X = 2) = \frac{5{,}460}{15{,}504} \approx 0.3522

Answer: The probability of drawing exactly 2 red cards is approximately

0.352

, or about 35.2%.

Visualization

Why It Matters

Quality control relies on this distribution: when an inspector pulls a sample from a finite lot of products, the hypergeometric model gives the exact probability of finding a certain number of defectives. It also underpins Fisher's exact test, a standard tool in biostatistics for analyzing contingency tables with small sample sizes.

Common Mistakes

Mistake: Using the binomial distribution instead when sampling without replacement from a small population.

Correction: The binomial assumes independence between draws (replacement). When the sample is a non-negligible fraction of the population (often cited as more than 5-10%), the hypergeometric distribution is the correct model because each draw changes the composition of the remaining pool.