Hypergeometric Distribution

Sampling without replacement

P(X = k) = C(K, k) · C(N − K, n − k) / C(N, n)

P(X = k)

0.27428

P(X ≤ k)

0.907233

P(X < k)

0.632953

P(X ≥ k)

0.367047

P(X > k)

0.0927671

Mean

1.25

Variance

0.863971

Std. deviation

0.9295

Mode

Support of k

0 – 5

About this calculator

The Hypergeometric Distribution Calculator computes the probability of drawing exactly k successes when sampling n items without replacement from a population of N that contains K successes. It returns the probability mass P(X = k), cumulative and tail probabilities, the mean, variance, standard deviation, and mode, and plots the full distribution. Use it for card and lottery odds, quality-control acceptance sampling, and statistics homework.

How to use the Hypergeometric Distribution calculator

Enter the values for your problem into the input fields.
Read the result — it updates instantly as you type.
Check the formula and the visual explanation to follow how the answer was found.
Copy the page URL to share the exact calculation.

Common examples

Deal 5 cards from a 52-card deck (N=52, K=13 hearts, n=5): P(exactly 2 hearts) ≈ 0.2743
A box of 20 parts has 4 defective (N=20, K=4, n=5): P(no defects, k=0) ≈ 0.2817
Lottery: pick 6 from 49 (N=49, K=6, n=6): P(match all 6, k=6) ≈ 7.15e-8
Capture-recapture: 50 tagged of 500 (N=500, K=50, n=30): mean tagged ≈ 3
Acceptance sampling: lot of 100 with 10 nonconforming (N=100, K=10, n=15): P(X ≤ 1) ≈ 0.5375

Frequently asked questions

How does the Hypergeometric Distribution calculator work?

Enter your values and the calculator applies the exact mathematical method for this problem, showing the result together with the formula it used. Everything is computed with high-precision arithmetic, so the answer you see is not limited by ordinary floating-point rounding.

When would I use the Hypergeometric Distribution calculator?

It is useful for homework and exam preparation, for checking work done by hand, and whenever a step in a larger problem needs this calculation done quickly and reliably.

How accurate are the results?

Calculations run at 30 significant digits of precision internally. Displayed values are rounded for readability, but the underlying result is far more precise than a typical hand or pocket-calculator computation.

Can I share a specific calculation with someone else?

Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same calculation for whoever clicks it.