Hypergeometric Distribution Table

Hypergeometric Distribution Table

The hypergeometric distribution models the probability of k successes in n draws from a finite population of size N containing K successes, without replacement.

Probability Formula

P(X = k) = [C(K,k) × C(N-K, n-k)] / C(N,n)

Where:

• N = population size
• K = number of successes in population
• n = number of draws (sample size)
• k = number of observed successes
• C(a,b) = binomial coefficient “a choose b”

Small Population Tables

N = 20, n = 5 (drawing 5 from population of 20)

N = 50, n = 10 (drawing 10 from population of 50)

Quality Control Application

Sampling from lot of N=100 items with D defectives

Sample size n = 10

Sample size n = 20

Cumulative Probabilities P(X ≤ k)

N = 50, K = 10, n = 10

Mean and Variance

Finite Population Correction

The factor (N-n)/(N-1) is the finite population correction, which reduces variance when sampling a large fraction of the population.

Comparison: Hypergeometric vs Binomial

Rule of thumb: Use binomial approximation when n < 0.05N (sample is less than 5% of population)

Example Problems

Example 1: Quality Control

Problem: A lot of 50 items contains 10 defectives. If 5 items are sampled, what’s the probability of exactly 2 defectives?

Solution:

• N = 50, K = 10, n = 5, k = 2
• P(X = 2) = [C(10,2) × C(40,3)] / C(50,5)
• P(X = 2) = 0.2098

Example 2: Card Drawing

Problem: In a deck of 52 cards (13 hearts), what’s P(exactly 3 hearts in 5 cards)?

Solution:

• N = 52, K = 13, n = 5, k = 3
• P(X = 3) = 0.0815

Example 3: Acceptance Sampling

Problem: A lot of 100 items has 15 defectives. Accept if sample of 10 has ≤ 1 defective. What’s P(accept)?

Solution:

• P(X ≤ 1) = P(X=0) + P(X=1)
• P(X ≤ 1) = 0.1821 + 0.3686 = 0.5507

Lottery/Selection Problems

Committee Selection

Problem: 8 men and 6 women available. Select committee of 5. P(exactly 2 women)?

Solution:

• N = 14, K = 6 (women), n = 5, k = 2
• P(X = 2) = [C(6,2) × C(8,3)] / C(14,5)
• P(X = 2) = 0.4196

Lottery Numbers

Problem: 49 numbers, 6 winning. Buy ticket with 6 numbers. P(matching exactly 3)?

Solution:

• N = 49, K = 6, n = 6, k = 3
• P(X = 3) = 0.0177 (about 1 in 57)

When to Use Hypergeometric

✓ Sampling WITHOUT replacement
✓ Finite population
✓ Sample size > 5% of population
✓ Quality control sampling
✓ Card/lottery problems
✓ Committee/selection problems

Related Distributions

Related Resources

• Hypergeometric Lesson - Complete guide
• Binomial Table - With replacement
• Probability Calculator - Calculate probabilities
• Combinations Calculator - Calculate C(n,k)