Hypergeometric Distribution
Print PageA Hypergeometric Distribution Calculator (also known as a Hypergeometric Probability Calculator, Sampling Without Replacement Utility, Finite Population Probability Analyzer, or Fisher’s Exact Test Probability Calculator) computes exact single-point probabilities (P(X = k) = [ C(K, k) · C(N - K, n - k) ] ÷ C(N, n)), lower cumulative probabilities (P(X ≤ k)), upper cumulative probabilities (P(X ≥ k)), expected mean (μ = n · [K ÷ N]), variance with finite population correction (σ2 = n · [K ÷ N] · [1 - K ÷ N] · [(N - n) ÷ (N - 1)]), and standard deviation for discrete sampling without replacement from a finite population.
Unlike the binomial distribution which assumes independent trials with replacement, the hypergeometric distribution models scenarios where each drawn item changes the probability of subsequent draws, such as lot quality acceptance sampling, card deck probabilities, and lottery draws.
Our free online Hypergeometric Distribution Calculator provides instant calculations across all finite population parameters:
- Probability Mass Function (PMF / P[X = k]):
P(X = k) = [ &binom;{K}{k} · &binom;{N - K}{n - k} ] ÷ &binom;{N}{n}. - Combination Formula (&binom;{n}{k}):
&binom;{n}{k} = n! ÷ [ k! · (n - k)! ]. - Expected Population Mean (μ):
μ = n · ( K ÷ N ). - Variance (σ2 with FPC):
σ2 = n · ( K ÷ N ) · ( 1 - K ÷ N ) · [ ( N - n ) ÷ ( N - 1 ) ]. - Finite Population Correction Factor (FPC):
FPC = ( N - n ) ÷ ( N - 1 ). - Hypergeometric vs. Binomial Approximation Threshold: When
n ≤ 0.05 · N(sample size ≤ 5% of population), Hypergeometric probabilities closely approximate BinomialB(n, p = K/N).
Master Hypergeometric Probability Reference Table (N = 100 Batch, K = 10 Defective, n = 10 Sampled)
The table below displays single-point probabilities P(X = k), lower cumulative probabilities P(X ≤ k), upper cumulative probabilities P(X ≥ k), and odds for a manufacturing quality audit drawing n = 10 components without replacement from a lot of N = 100 containing K = 10 defective units:
| Sample Success Count (k Defective Units) | PMF Combination Formula | Exact P(X = k) | Lower Cumulative P(X ≤ k) | Upper Cumulative P(X ≥ k) | Acceptance Quality Audit Status |
|---|---|---|---|---|---|
| k = 0 Defective Units | [&binom;{10}{0} · &binom;{90}{10}] ÷ &binom;{100}{10} | 33.0476% (0.3305) | 33.0476% | 100.0000% | Pristine Clean Batch Sample |
| k = 1 Defective (μ Expected Mean) | [&binom;{10}{1} · &binom;{90}{9}] ÷ &binom;{100}{10} | 41.3094% (0.4131) | 74.3570% (74.36%) | 66.9524% | Most Likely Single-Outcome Peak |
| k = 2 Defective Units | [&binom;{10}{2} · &binom;{90}{8}] ÷ &binom;{100}{10} | 19.3340% (0.1933) | 93.6910% (93.69%) | 25.6430% | Acceptable Quality Limit (AQL) |
| k = 3 Defective Units | [&binom;{10}{3} · &binom;{90}{7}] ÷ &binom;{100}{10} | 5.1090% (0.0511) | 98.8000% | 6.3090% | Elevated Defect Concern |
| k ≥ 4 Defective Units | ∑ P(X = 4..10) | 1.2000% (0.0120) | 100.0000% | 1.2000% | Batch Rejection Threshold (>98.8% unusual) |
Step-by-Step Batch Quality Control Inspection Calculation
To evaluate a quality assurance audit inspecting n = 10 parts without replacement from a box of N = 100 total parts containing K = 10 defective units:
Step 1 (Calculate Total Ways to Choose Sample &binom;{100}{10}): &binom;{100}{10} = 17,310,309,456,440 combinations
Step 2 (Calculate Expected Defective Count μ): μ = n · (K ÷ N) = 10 · (10 ÷ 100) = 1.0 defective part
Step 3 (Calculate Variance σ^2 with FPC): σ^2 = 10 · 0.10 · 0.90 · (90 ÷ 99) = 0.90 · 0.90909 = 0.81818 &implies; σ = 0.9045
Step 4 (Calculate Exact 0 Defective Probability P[X = 0]): P(X = 0) = [ &binom;{10}{0} · &binom;{90}{10} ] ÷ &binom;{100}{10} = [ 1 · 5,720,645,481,906 ] ÷ 17,310,309,456,440 = 0.33048 ≈ 33.05%
Step 5 (Calculate Probability of At Most 2 Defectives P[X ≤ 2]): P(X ≤ 2) = 33.05% + 41.31% + 19.33% = 93.69%
Thus, the inspector expects 1 defective component on average, with a 93.69% probability of finding 2 or fewer defectives in the sample.
Sampling Distributions Comparison: Hypergeometric vs. Binomial vs. Fisher’s Exact Test
Below is a comparative reference chart detailing when to use the Hypergeometric distribution versus related sampling models:
| Probability Model | Replacement Condition | Trial Independence | Primary Practical Application |
|---|---|---|---|
| Hypergeometric Distribution | WITHOUT Replacement (Finite N) | DEPENDENT Trials (Probabilities change each draw) | Lot quality inspection, card hands, lottery audits. |
| Binomial Distribution | WITH Replacement (or Infinite N) | INDEPENDENT Trials (Constant p each draw) | Coin flips, website conversion rates, n < 5% N. |
| Fisher’s Exact Test | 2×2 Contingency Table Marginals Fixed | Hypergeometric Conditional Distribution | Small-sample 2×2 categorical significance tests. |
History & Mathematics: 1713 Jacob Bernoulli to 1935 Ronald Fisher
1713 Jacob Bernoulli & Card Hand Probabilities
In 1713, Swiss mathematician Jacob Bernoulli analyzed early card sampling games of chance without replacement in Ars Conjectandi.
1935 Ronald A. Fisher & Fisher’s Exact Test
In 1935, legendary statistician Ronald A. Fisher published The Design of Experiments, establishing Fisher’s Exact Test based directly on the hypergeometric distribution for evaluating 2×2 contingency tables in small-sample research.
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Frequently Asked Questions (FAQ)
What is the Hypergeometric Distribution formula?
The single-point PMF formula is P(X = k) = [ &binom;{K}{k} · &binom;{N-K}{n-k} ] ÷ &binom;{N}{n}.
What is the difference between Hypergeometric and Binomial distributions?
The Hypergeometric distribution models sampling without replacement from a finite population (dependent trials), whereas the Binomial distribution models sampling with replacement (independent trials).
When can you use the Binomial distribution to approximate the Hypergeometric distribution?
You can safely use the Binomial approximation whenever the sample size n is less than or equal to 5% of the total population size (n ≤ 0.05 · N).