Home » Statistics & Probability Calculators » Hypergeometric Distribution Calculator

Hypergeometric Distribution Calculator

Home › Statistics & Probability Calculators › Hypergeometric Distribution Calculator

Hypergeometric Distribution Calculator

The hypergeometric distribution models the number of successes when you draw n items without replacement from a population of N items that contains K successes: the number of aces in a five-card hand, the number of defective parts in a sample from a batch, the number of women on a randomly chosen committee. Enter N, K, n and the value(s) of k.

The calculator applies P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n), shows the three binomial coefficients with their values, applies the cumulative rule you chose and lists a probability table over the possible range of k (which is limited on both sides when n is large). The mean nK/N and the variance with its finite population factor are reported.

When the population is much larger than the sample (N more than about 20 times n) the draws are nearly independent and the binomial distribution with p = K/N is a good approximation.

Result and step-by-step solution

P(X ≤ 3)
0.82094

Drawing 10 without replacement from 50 items of which 12 are successes. Mean nK/N = 2.4, SD = 1.2202

012345678910
  1. Formula
    P(X = k) = C(K, k) × C(N − K, n − k) ÷ C(N, n)
  2. Single term at k = 3
    P(X = 3) = C(12, 3) × C(38, 7) ÷ C(50, 10) = 220 × 12620256 ÷ 10272278170 = 0.27029
  3. Apply the rule
    P(X ≤ 3) = 0.82094
  4. Probability table (possible k from 0 to 10)
    kP(X = k)P(X ≤ k)P(X ≥ k)
    00.046020.046021
    10.190430.236450.95398
    20.314210.550660.76355
    30.270290.820940.44934
    40.133030.953980.17906
    50.03870.992680.04602
    60.006640.999310.00732
    70.000650.999970.00069
    80.0000310.00003
    98.138 × 10^−718.203 × 10^−7
    106.425 × 10^−916.425 × 10^−9

Frequently asked questions

What is the difference between binomial and hypergeometric?
Binomial draws with replacement (or from an infinite population), so p stays constant. Hypergeometric draws without replacement, so each draw changes the odds.
What is the probability of exactly 2 aces in 5 cards?
N = 52, K = 4, n = 5, k = 2: C(4,2) × C(48,3) ÷ C(52,5) = 6 × 17296 ÷ 2598960 = 0.0399.
Why can't k be any value from 0 to n?
You cannot draw more successes than exist (k ≤ K) or more failures than exist (n − k ≤ N − K). The table shows only the possible values.
How do the parameters map to a TI-84 or Excel?
Excel’s HYPGEOM.DIST(k, n, K, N) uses the same four quantities. SciPy’s hypergeom(M, n, N) uses M for the population, n for successes and N for draws.

Related calculators

Last reviewed: September 25, 2026. Calculations run in your browser and were validated against SciPy.

Scroll to Top