Hypergeometric Distribution Calculator
Calculate probabilities for sampling without replacement from finite populations
Distribution Parameters
Total number of items in the population
Number of items with desired feature in population
Number of items drawn from population
Number of success items to find in sample
Results
Formula: P(X = k) = C(K,k) × C(N-K,n-k) / C(N,n)
Parameters: N=48, K=12, n=10, k=4
Mean formula: μ = n × K / N = 10 × 12 / 48 = 2.5000
Chocolate Bar Example
Problem Setup
Scenario: Bag contains 12 dark and 36 white chocolate bars
Population size (N): 48 total bars
Success states (K): 12 dark chocolate bars
Sample size (n): 10 bars drawn without replacement
Question: What's the probability of exactly 4 dark bars?
Solution
P(X = 4) = C(12,4) × C(36,6) / C(48,10)
P(X = 4) = 495 × 1,947,792 / 6,540,715,896
P(X = 4) ≈ 0.1474 or 14.74%
Expected value: μ = 10 × 12 / 48 = 2.5 dark bars
Key Characteristics
Finite Population
Fixed number of items total
No Replacement
Items not returned to population
Binary Outcomes
Success or failure states only
Fixed Sample
Predetermined number of draws
When to Use
Quality control sampling
Card game probabilities
Fisher's exact test
Ecological sampling
Medical testing accuracy
Understanding Hypergeometric Distribution
What is Hypergeometric Distribution?
The hypergeometric distribution describes the probability of getting exactly k successes in n draws from a finite population of size N containing K success items, without replacement.
Key Properties
- •Mean: μ = n × K / N
- •Variance: σ² = n × (K/N) × (1-K/N) × (N-n)/(N-1)
- •Maximum k value is min(n, K)
- •Becomes binomial when N is very large
Probability Mass Function
P(X = k) = C(K,k) × C(N-K,n-k) / C(N,n)
- N: Population size
- K: Number of success items in population
- n: Sample size (number of draws)
- k: Number of successes in sample
- C(n,k): Binomial coefficient ("n choose k")
Note: All draws must be without replacement for this distribution to apply.
Hypergeometric vs. Binomial Distribution
Hypergeometric
- • Sampling without replacement
- • Finite population
- • Probability changes with each draw
- • Used for small populations
Binomial
- • Sampling with replacement
- • Infinite or very large population
- • Constant probability
- • Used for independent trials