Calculator
Compute exact and cumulative hypergeometric probabilities for sampling without replacement.
Valid support: max(0, n - (N - K)) ≤ X ≤ min(n, K)
What is the hypergeometric distribution?
The hypergeometric distribution models the probability of getting a specific number of successes when you draw a sample from a finite population without replacement. That “without replacement” part is the key difference from a binomial model.
In plain terms: if you take items out and do not put them back, the odds change from draw to draw. Hypergeometric probability captures that changing-odds process exactly.
When should you use this calculator?
- Quality control: probability of defective parts in a random inspection sample.
- Card games: chance of drawing a certain number of hearts, aces, or other target cards.
- Audit sampling: probability of selecting a number of problematic records.
- Biology and genetics: overlap/enrichment style calculations from finite sets.
- Lot acceptance decisions in manufacturing and procurement.
Parameter meanings
N: Population size
Total number of items in the full population.
K: Number of successes in the population
How many of those N items are classified as “success” (or “target”).
n: Sample size (draws)
How many items you draw from the population without replacement.
k: Number of observed successes in the sample
The count of target items found in your sample. Depending on calculator mode, this may be exact, a threshold, or a range endpoint.
Formula used
For exact probability of observing exactly k successes:
P(X = k) = [C(K, k) × C(N − K, n − k)] / C(N, n)
where C(a, b) is the combination function (“a choose b”). The calculator also computes cumulative probabilities by summing valid exact probabilities over the requested range.
How to use the calculator
- Select the probability type: exact, at most, at least, or between.
- Enter N, K, and n as integers.
- Enter k (and k2 if using “between”).
- Click Calculate.
The result panel shows:
- The probability expression evaluated.
- Probability in decimal and percent form.
- Distribution support bounds.
- Expected value and variance for quick interpretation.
Hypergeometric vs. binomial: quick comparison
Hypergeometric
- Sampling without replacement.
- Finite population with changing draw probabilities.
- Exact for finite-lot scenarios.
Binomial
- Sampling with replacement (or effectively infinite population).
- Constant success probability each trial.
- Often used as approximation when population is very large relative to sample.
Worked mini example
Suppose a deck has 52 cards, with 4 aces. You draw 5 cards. What is the probability of getting exactly 1 ace?
- N = 52
- K = 4
- n = 5
- k = 1
Enter those values with mode P(X = k) and calculate. This is a classic hypergeometric setup because cards are not replaced during the hand draw.
Common input mistakes to avoid
- Using decimals instead of integers.
- Setting K greater than N.
- Setting n greater than N.
- For range mode, entering k1 greater than k2.
- Using binomial assumptions for without-replacement scenarios.
Final thoughts
A hypergeometric distribution calculator is essential whenever you care about realistic finite-population sampling. If your process draws from a fixed pool and each draw changes what remains, this model gives the right probability framework.