Hypergeometric Distribution Calculator
Draws without replacement — cards, lotteries, quality samples.
Enter your values
Enter the population size n you are working with.
Enter the successes in the population k you are working with.
Enter the sample drawn n you are working with.
Enter the successes wanted k you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- 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)
At a glance
| What it does | Draws without replacement — cards, lotteries, quality samples. |
|---|---|
| Category | Statistics |
| Inputs needed | Population size N, Successes in the population K, Sample drawn n, Successes wanted k |
| Main output | P(exactly k) |
| Formula | P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n) |
| Cost | Free — no sign-up, no download |
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.
Inputs explained
- Population size N
- Enter the population size n you are working with.
- Successes in the population K
- Enter the successes in the population k you are working with.
- Sample drawn n
- Enter the sample drawn n you are working with.
- Successes wanted k
- Enter the successes wanted k you are working with.
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.
What do I need to enter into the Hypergeometric Distribution Calculator?
Just 4 values: population size n, successes in the population k, sample drawn n, successes wanted k. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Hypergeometric Distribution Calculator work out the answer?
The hypergeometric distribution models sampling without replacement: P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n). It applies the formula P(X = k) = C(K,k)·C(N−K,n−k) ÷ C(N,n) and shows the working so you can check each step by hand.
Is the Hypergeometric Distribution Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.