Hypergeometric Distribution

Probability Distributions

Sampling Without Replacement — Finite Populations

The hypergeometric distribution models the number of successes when sampling without replacement from a finite population. Every draw changes the odds.

Card games — probability of drawing aces in a poker hand
Quality inspection — defective items in a batch sample
Ecology — capture-recapture population estimation
Legal sampling — random drug testing from a workforce

Unlike the binomial, the hypergeometric accounts for the fact that each draw depletes the population.

Core Concepts

The hypergeometric distribution models the number of successes when sampling without replacement from a finite population. It is the correct distribution for card games, quality inspection of small batches, and any scenario where the population is depleted as items are drawn.

Mean and Variance

The Finite Population Correction Factor

Convergence to Binomial

Symmetry Property

Worked Example: Lottery / Card Draw

Python Implementation

import numpy as np
from scipy import stats

np.random.seed(42)

# Hypergeometric parameters
N, K, n = 52, 4, 5  # deck, aces, draws

# Theoretical mean and variance
mean_theory = n * K / N
var_theory = n * (K/N) * (1 - K/N) * (N - n) / (N - 1)
print(f"Hyp({N}, {K}, {n}):")
print(f"  Theoretical mean: {mean_theory:.4f}")
print(f"  Theoretical variance: {var_theory:.4f}")

# Simulate
n_sims = 100000
samples = np.random.hypergeometric(K, N - K, n, size=n_sims)
print(f"  Empirical mean:   {np.mean(samples):.4f}")
print(f"  Empirical variance: {np.var(samples, ddof=0):.4f}")

# Compare with binomial (with replacement)
binom_var = n * (K/N) * (1 - K/N)
fpc = (N - n) / (N - 1)
print(f"\n  Binomial variance (with replacement): {binom_var:.4f}")
print(f"  Finite population correction: {fpc:.4f}")
print(f"  Hyp var / Bin var = {var_theory / binom_var:.4f}  (should be {fpc:.4f})")

Python Implementation: Batch Quality Inspection

import numpy as np
from scipy import stats

np.random.seed(42)

# Quality control: batch of 100 items, 8 defective, sample 10
N, K, n = 100, 8, 10
samples = np.random.hypergeometric(K, N - K, n, size=50000)

print(f"Quality Inspection: N={N}, K={K} defective, sample n={n}")
print(f"PMF values:")
for k in range(min(n, K) + 1):
    pmf = stats.hypergeom.pmf(k, N, K, n)
    print(f"  P(X={k}) = {pmf:.4f}")

print(f"\nSimulated mean: {np.mean(samples):.4f} (theoretical: {n*K/N:.4f})")

# Probability of finding at least 2 defectives
prob_ge_2 = 1 - stats.hypergeom.cdf(1, N, K, n)
print(f"P(X >= 2 defectives in sample) = {prob_ge_2:.4f}")

Hypergeometric Distribution — Sampling Without Replacement