Hypergeometric probability
Find successes when sampling without replacement from a finite population.
Hypergeometric probability is one of 7 distributions formulas in the probability section of this library, and it is used at ap · university level.
Why hypergeometric probability works
Count the ways to pick k items from the successes and the remaining picks from the failures, then divide by the number of ways to pick the sample from the whole population. Because every sample of that size is equally likely, counting the favourable ones and dividing is all the probability there is.
What each symbol means
$N$ is population size, $K$ successes, $n$ draws, and $k$ observed successes.
Hypergeometric probability: when it holds
Selections are without replacement and all size-$n$ samples are equally likely.
When it stops applying
It assumes drawing without replacement from a fixed finite population where every sample of the given size is equally likely. When the sample is a tiny slice of a huge population the difference disappears: the same problem scaled to 10000 items gives 0.28801, which matches the binomial answer of 0.288 to three decimal places.
Hypergeometric probability: a worked example
From $N=10,K=4,n=3$, exactly $k=2$ has probability $\binom42\binom61/\binom{10}3$.
The mistake to avoid
What people do: Using the binomial with a success probability of the population share.
Why it goes wrong: From 10 items with 4 successes, drawing 3 and getting exactly 2 has probability 0.3. The binomial with probability 0.4 gives 0.288, because it pretends each draw is taken from a full population.
Do this instead: Check whether the item is replaced. If it is not, the population shrinks with every draw and this counting formula is the right one.
Hypergeometric probability: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $N$ is population size, $K$ successes, $n$ draws, and $k$ observed successes.
- Check the conditions before substituting. Selections are without replacement and all size-$n$ samples are equally likely.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most probability slips.
Where this formula fits
- Subject
- Probability formulas — 17 entries in this library
- Topic
- Distributions
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where hypergeometric probability comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Probability Foundations — the lesson behind this formula: model outcomes, complements, and conditional information.
- Statistics Calculator — check your substitution and the value it produces.
- Study probability — the subject guide that explains the ideas these formulas compress.
- Statistics Practice — questions that make you retrieve the formula instead of recognising it.
- All 17 probability formulas — the full grouped reference, or the complete formula library.
Questions about hypergeometric probability
When is the binomial close enough to use instead?
The usual guideline is when the sample is under about 5 percent of the population. Beyond that, the shrinking pool matters and the two answers separate noticeably.
What is the mean of this distribution?
It is the number of draws times the population share of successes, the same as the binomial mean. Only the variance differs, being smaller by a correction factor for the finite population.
Is this the formula behind lottery odds?
Yes. Matching 3 of the 5 drawn numbers when choosing from 50 has probability about 0.00467, computed exactly this way with the matched and unmatched numbers counted separately.
What happens if I ask for more successes than the population contains?
The probability is zero, and the formula produces it automatically, because there is no way to choose more items than exist and that combination evaluates to zero.