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
1
Support of k
0 – 5

The most likely number of successes is 1; the chart shows the probability of each outcome from 0 to 5.

Hypergeometric probability mass functionHypergeometric probability mass function012345

关于这个计算器

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.

常见示例

  • 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