Hypergeometric Distribution Calculator
Calculate sampling without replacement probability ($P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}}$), expected mean, and finite population variance.
Calculate Sampling Without Replacement Probabilities
Enter population size (N), population successes (K), sample size (n), and sample successes (k).
Calculation Results
Calculated using hypergeometric probability mass function: P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}
Quick Summary
The Hypergeometric Distribution Calculator evaluates exact sampling-without-replacement probability $P(X = k)$, cumulative probabilities, expected mean ($n K / N$), and finite population variance.
Formula Explanation
P(X = k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}}
E(X) = n \cdot \frac{K}{N}, \quad \text{Var}(X) = n \cdot \frac{K}{N} \cdot \left(1 - \frac{K}{N}\right) \cdot \left(\frac{N - n}{N - 1}\right)How It Works
Unlike binomial distribution which assumes replacement and constant probability, hypergeometric distribution models drawing items without replacement from a finite population $N$. The Hypergeometric Distribution Calculator evaluates exact combinatorial probabilities $\binom{K}{k}$.
Step-by-Step Worked Example
Practical Problem: A lot of $N = 50$ manufactured parts contains $K = 10$ defective items. A inspector draws $n = 5$ parts without replacement. Calculate probability of drawing exactly $k = 2$ defective items.
- Step 1: Calculate ways to choose $k = 2$ defects from $K = 10$: $\binom{10}{2} = \mathbf{45}$.
- Step 2: Calculate ways to choose remaining $n - k = 3$ good parts from $N - K = 40$: $\binom{40}{3} = \mathbf{9,880}$.
- Step 3: Calculate total ways to choose sample of $n = 5$ from $N = 50$: $\binom{50}{5} = \mathbf{2,118,760}$.
- Step 4: Evaluate hypergeometric PMF: $P(X = 2) = \frac{45 \times 9,880}{2,118,760} = \frac{444,600}{2,118,760} = \mathbf{0.209840\text{ (20.98\%)}}.$
- Step 5: Calculate Expected Mean: $E(X) = 5 \times (10 / 50) = \mathbf{1.0000\text{ defect expected}}$.
Real-World Calculation Examples
Scenario 1: Quality Control Lot (N = 50, K = 10, n = 5, k = 2)
Parameters: N = 50, K = 10, n = 5, k = 2
Result: P(X = 2) = 0.2098 (20.98% chance).
Scenario 2: Card Game Poker Hand (N = 52, K = 4 Aces, n = 5)
Parameters: N = 52, K = 4, n = 5, k = 2
Result: P(X = 2) = 0.0399 (3.99% pair of aces probability).
Scenario 3: Lottery Ticket Ball Selection
Parameters: N = 49 total balls, K = 6 winning numbers, n = 6 drawn
Result: Hypergeometric jackpot probability.
Scenario 4: Jury Selection Demographic Representation
Parameters: N = 100 pool, K = 40 group members, n = 12 jurors
Result: Expected representation E(X) = 4.8 jurors.
Key Benefits of Using This Calculator
Sampling Without Replacement
Calculates true probabilities when items are selected without replacement from finite populations.
Finite Population Correction (FPC)
Variance includes exact FPC adjustment factor $\frac{N - n}{N - 1}$.
Exact & Cumulative Probabilities
Outputs $P(X = k)$, $P(X \le k)$, and $P(X \ge k)$.
100% Free & Client-Side
Executes locally in your browser with zero latency or web server transmission.
Frequently Asked Questions (FAQ)
What is a hypergeometric distribution?
The hypergeometric distribution models the number of successes $k$ in a sample of size $n$ drawn without replacement from a finite population $N$ containing $K$ successes.
What is difference between hypergeometric and binomial distribution?
Binomial distribution samples with replacement (constant $p$); hypergeometric distribution samples without replacement (changing probabilities).
When does hypergeometric distribution approach binomial distribution?
When sample size $n$ is very small relative to population size $N$ ($n/N < 0.05$), hypergeometric probabilities converge to binomial probabilities.
How do I calculate hypergeometric probabilities in Excel?
Use formula =HYPGEOM.DIST(k, n, K, N, FALSE) for exact $P(X=k)$ or TRUE for cumulative $P(X \le k)$.
What is Fisher's Exact Test?
Fisher's Exact Test uses the hypergeometric distribution to calculate exact $p$-values for $2 \times 2$ contingency tables with small sample sizes.
What is formula for hypergeometric mean?
$E(X) = n \cdot \frac{K}{N}$.
What is Finite Population Correction factor in hypergeometric variance?
The FPC multiplier is $\frac{N - n}{N - 1}$, which reduces variance as sample size $n$ approaches population size $N$.
What happens if sample size n equals population size N?
If $n = N$, all population items are selected, so $k = K$ with $P(X = K) = 1.0$ and variance $= 0$.
What are support boundaries for k?
Success count $k$ must satisfy $\max(0, n - (N - K)) \le k \le \min(n, K)$.
Why is hypergeometric distribution important in manufacturing acceptance sampling?
Quality managers use hypergeometric distribution to determine lot acceptance probabilities when inspecting sample parts without replacement.