Hypergeometric Distribution Calculator

What is Hypergeometric Distribution?

The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K success states. It differs from the binomial distribution in that sampling is done without replacement.

Hypergeometric Distribution Properties

Hypergeometric Distribution Formulas

1. Probability Mass Function (PMF)

Probability of exactly k successes:

P(X = k) = [C(K, k) × C(N-K, n-k)] / C(N, n)
where C(a, b) = a! / [b! × (a-b)!]
Constraints: max(0, n+K-N) ≤ k ≤ min(n, K)

2. Cumulative Distribution Function (CDF)

Probability of at most k successes:

P(X ≤ k) = Σ P(X = i) for i = 0 to k
P(X ≥ k) = 1 - P(X ≤ k-1)
P(a ≤ X ≤ b) = Σ P(X = i) for i = a to b

3. Mean and Variance

Distribution moments:

• Mean: μ = n × (K/N)
• Variance: σ² = n × (K/N) × ((N-K)/N) × ((N-n)/(N-1))
• Standard Deviation: σ = √σ²
• The factor (N-n)/(N-1) is the finite population correction

4. Binomial Approximation

When N is large relative to n:

Hypergeometric ≈ Binomial if n/N < 0.05
Binomial parameters: p = K/N
P_bin(X=k) = C(n,k) × pᵏ × (1-p)ⁿ⁻ᵏ

Real-World Applications

Quality Control & Manufacturing

• Defective items: Probability of finding defective products in a sample
• Acceptance sampling: Determine if a batch should be accepted or rejected
• Inventory management: Probability of stockouts or excess inventory
• Six Sigma: Process capability analysis

Gambling & Games

• Lottery probabilities: Chance of matching numbers in lottery draws
• Card games: Probability of getting specific cards in poker, bridge
• Bingo: Probability of completing patterns
• Raffles: Chance of winning with multiple tickets

Biology & Ecology

• Capture-recapture: Estimating population sizes in ecology
• Genetic sampling: Probability of specific genotypes in a sample
• Species diversity: Probability of finding rare species in samples
• Medical testing: Probability of disease in screened populations

Social Sciences & Business

• Survey sampling: Probability of specific responses in surveys
• Auditing: Probability of finding errors in financial audits
• Political polling: Probability of specific voting patterns
• Market research: Sampling consumer preferences

Comparison with Binomial Distribution

Step-by-Step Calculation Example

Example: Quality Control Inspection

Scenario: A batch of 100 items contains 10 defectives. An inspector randomly selects 20 items without replacement. What is the probability that exactly 3 items are defective?

1. Identify parameters:
   • N = 100 (population size)
   • K = 10 (number of defectives in population)
   • n = 20 (sample size)
   • k = 3 (desired successes in sample)
2. Check constraints:
   • k ≤ K: 3 ≤ 10 ✓
   • k ≤ n: 3 ≤ 20 ✓
   • n-k ≤ N-K: 20-3=17 ≤ 100-10=90 ✓
3. Calculate combinations:
   • C(K, k) = C(10, 3) = 10!/(3!×7!) = 120
   • C(N-K, n-k) = C(90, 17) = 90!/(17!×73!) ≈ 1.146×10¹⁸
   • C(N, n) = C(100, 20) = 100!/(20!×80!) ≈ 5.360×10²⁰
4. Calculate probability:
   • P(X=3) = [C(10,3) × C(90,17)] / C(100,20)
   • P(X=3) = [120 × 1.146×10¹⁸] / 5.360×10²⁰
   • P(X=3) ≈ 0.2566
5. Interpretation: There is approximately a 25.66% chance that exactly 3 out of 20 randomly selected items will be defective.

Common Hypergeometric Problems

Related Calculators

Frequently Asked Questions (FAQs)

Q: When should I use hypergeometric instead of binomial distribution?

A: Use hypergeometric when sampling is done WITHOUT replacement from a finite population. Use binomial when sampling is WITH replacement, or when the population is very large relative to the sample size (n/N < 0.05).

Q: What is the finite population correction factor?

A: The factor (N-n)/(N-1) in the variance formula accounts for sampling without replacement. It reduces the variance compared to binomial distribution. When N is very large, this factor approaches 1.

Q: Can k be greater than K or n in hypergeometric distribution?

A: No, k must satisfy: max(0, n+K-N) ≤ k ≤ min(n, K). These constraints ensure all combinations are valid (can't draw more successes than exist, etc.).

Q: How does hypergeometric distribution relate to Fisher's exact test?

A: Fisher's exact test uses the hypergeometric distribution to calculate the exact probability of observing a particular arrangement of data in a 2×2 contingency table, useful for small sample sizes.

Calculate exact probabilities for sampling without replacement with Toolivaa's free Hypergeometric Distribution Calculator, and explore more statistical tools in our Math Calculators collection.