Find the probability of drawing exactly k successes in n draws without replacement from a finite population — the classic “cards from a deck” probability problem.
How it works
The formula is P(X=k) = C(K,k) × C(N−K,n−k) ÷ C(N,n). Drawing exactly 2 hearts in a 5-card hand from a standard 52-card deck (13 hearts) has a probability of about 27.4%.
What this does not include
This is distinct from this site’s binomial distribution calculator, which assumes draws with replacement (or an effectively infinite population) — a different sampling model that gives different results, especially with a small population.
How to use this calculator
- Enter the total population size (N).
- Enter how many “success” items exist in the population (K).
- Enter your sample size (n).
- Enter the number of successes you want the probability of (k).
Why does drawing without replacement need a different formula than the binomial?
Without replacement, each draw changes the composition of what remains, so the probability of success shifts after every draw — the binomial distribution assumes that probability stays constant throughout.
When does the hypergeometric distribution approximate the binomial?
When the population is very large relative to the sample size, removing a few items barely changes the remaining proportions, so the two distributions converge.
What’s a classic real-world example of this distribution?
Card games are the textbook case — the probability of drawing a specific number of a particular suit or rank from a shuffled deck follows exactly this distribution.
