What this calculator does
The hypergeometric distribution covers sampling without replacement from a finite population. Drawing 8 items from a population of 50 containing 10 successes, the chance of getting exactly 3 is 14.7%.
The distinction from the binomial is that each draw changes what remains. Take a success out of the pool and the next draw is less likely to be one. This matters whenever the sample is a meaningful fraction of the population; when the population is very large relative to the sample, the two distributions converge and the simpler binomial is fine.
The formula
The probability is the number of ways to choose k successes from the K available, multiplied by the ways to choose the remaining n−k items from the N−K failures, divided by the total ways to choose n items from N. The mean is n times K over N, which matches the binomial, but the variance is smaller by the finite population correction factor.
| Term | Meaning |
|---|---|
| N | Population size. |
| K | Number of successes in the population. |
| n | Sample size drawn. |
| Without replacement | Items are not returned, so each draw changes the remaining composition. |
The inputs explained
| Field | What to enter |
|---|---|
| Population size (N) | Total population size. |
| Successes in population (K) | How many of the population are successes. Cannot exceed N. |
| Sample size (n) | How many items you draw. Cannot exceed N. |
| Successes wanted in sample (k) | Successes wanted in the sample. Cannot exceed K or n. |
When to use it
Acceptance sampling
Inspecting a sample from a finite batch, where drawing items changes what is left in the batch.
Card and lottery problems
Any draw from a deck or pool without replacement is hypergeometric rather than binomial.
Auditing a finite set of records
Sampling 8 of 50 files is a substantial share of the population, so the correction matters.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What are the chances drawing 8 from 50?
The chance of each number of successes in the sample.
| Successes in sample (k) | P(exactly k successes) | Expected successes (mean) | Standard deviation |
|---|---|---|---|
| k = 0 | 14.3% | 1.60 | 1.05 |
| k = 1 | 34.7% | 1.60 | 1.05 |
| k = 2 | 32.2% | 1.60 | 1.05 |
| k = 3 | 14.7% | 1.60 | 1.05 |
| k = 5 | 0.464% | 1.60 | 1.05 |
Questions
When should I use hypergeometric instead of binomial?
Whenever sampling without replacement from a population small enough that the draws affect each other. A common rule is to use hypergeometric when the sample exceeds about 5% of the population; below that the binomial approximation is close enough.
Why is the variance smaller than binomial?
Because sampling without replacement is self-correcting. Drawing several successes early leaves fewer available, which pulls later draws back toward the average. That constraint reduces variability, which is exactly what the finite population correction factor expresses.
Is the mean the same as binomial?
Yes. The expected number of successes is n times K over N either way, since on average each draw has the same chance of being a success. Only the spread differs, not the centre.
What happens as the population grows?
The hypergeometric converges to the binomial. With a very large population, removing a few items barely changes the composition, so the draws become effectively independent and the two distributions become indistinguishable.
For sampling with replacement, see the binomial distribution calculator. For lottery-style draws, see the lottery odds calculator.