Hypergeometric Distribution Calculator
Draws without replacement — cards, lotteries, quality samples.
Enter your values
ⓘ Live results
Results
- P(exactly k)
- 27.428%
- P(at most k)
- 90.7233%
- P(at least k)
- 36.7047%
- Expected successes
- 1.25
- Standard deviation
- 0.93
Quick answer
The hypergeometric distribution models sampling without replacement: P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n).
P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n)
How to use the Hypergeometric Distribution Calculator
- 1Enter the total population and how many of them count as successes.
- 2Enter how many items you draw.
- 3Enter the number of successes you want in the draw.
Worked example
Using the values the calculator loads with:
Inputs
- Population size N52
- Successes in the population K13
- Sample drawn n5
- Successes wanted k2
Results
- P(exactly k)27.428%
- P(at most k)90.7233%
- P(at least k)36.7047%
- Expected successes1.25
- Standard deviation0.93
Frequently asked questions
When should I use binomial instead?
When you replace each item, or when the sample is under about 5% of the population — then the two answers barely differ.
Can it handle lottery odds?
Yes. Set N to the ball count, K to the numbers on your ticket, n to the balls drawn and k to the matches.
About this calculator
- Built and reviewed by James Whitfield · How we build and test calculators
- Report an error
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.