Hypergeometric Distribution

The Hypergeometric distribution models the number of successes when you sample without replacement from a finite population.

Setup

• Population size: $N$
• Number of successes in population: $K$
• Draws (sample size): $n$
• Random variable: $X$ = number of successes in the sample

Then:

PMF

Intuition

You’re counting: “choose $k$ successes” and “choose $n-k$ failures”, divided by “choose $n$ total”.

Hypergeometric vs Binomial

• Hypergeometric: without replacement → probabilities change after each draw.
• Binomial: with replacement / independent trials → probability stays constant.

If the population is huge relative to the sample (e.g., $n \ll N$), Hypergeometric is often well-approximated by a Binomial with $p=K/N$.

Key facts

• Mean: $E[X] = n\frac{K}{N}$
• Variance:

The factor $\frac{N-n}{N-1}$ is the finite population correction (variance is smaller than Binomial because you’re not sampling independently).

When to use

• Cards without replacement
• Auditing / QA sampling from a finite batch
• Any “draw $n$ items from $N$ without putting them back” scenario

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