Hypergeometric Calculator - Sampling Without Replacement
Use this hypergeometric calculator to solve sampling without replacement problems. Enter N, K, n, and k to get exact and cumulative probabilities.
Hypergeometric Calculator
Results
What is a Hypergeometric Calculator?
A hypergeometric calculator finds the probability of drawing a certain number of successes from a finite population when every draw happens without replacement. It is the right tool when each selected item changes what remains for the next draw.
- •Card probability, such as drawing aces, hearts, or face cards from a fixed hand size.
- •Quality control sampling, such as estimating the chance that inspected items include defects.
- •Committee selection, classroom examples, lottery-style draws, and inventory audits.
This hypergeometric probability calculator is different from a general odds tool because it respects the finite population. To review broader probability rules, use our Probability Calculator to compare event likelihoods.
How Hypergeometric Calculation Works
The hypergeometric distribution formula counts favorable samples and divides by all possible samples of the same size. It uses combinations because draw order does not matter.
N is the population size, K is the number of successes in that population, n is the sample size, and k is the success count you want to test. The calculator also sums exact probabilities when you choose at most, at least, greater than, fewer than, or between mode.
According to Encyclopaedia Britannica, the hypergeometric distribution applies when selections are made from two groups without replacement.
To inspect the combination part of the formula, use our Permutation and Combination Calculator to calculate C(n,r) values directly.
Key Hypergeometric Concepts
Population Size (N)
The total number of items that could be selected. A deck of cards has N = 52; a shipment might have N = 1,000 units.
Success States (K)
The number of population items that count as the outcome you care about, such as aces, defective parts, or eligible members.
Sample Size (n)
The number of items drawn without replacement. Once an item is selected, it is not returned before the next draw.
Hypergeometric vs Binomial
Use hypergeometric when draws are dependent; use binomial when each trial has the same independent probability.
For a related discrete event model based on a fixed average rate, explore our Poisson Distribution Calculator to model counts over time or space.
How to Use This Calculator
Enter N
Use the total population size, such as 52 cards or 100 inspected units.
Enter K
Enter the number of population items that count as successes.
Enter n and k
Add the sample size and target success count you want to test.
Choose Mode
Select exactly, at most, at least, fewer than, greater than, or between mode.
If you are planning a study before collecting data, use our Sample Size Calculator to estimate how many observations you need.
Benefits of Using This Calculator
- •Correct finite sampling: Avoid replacement mistakes by using the model built for dependent draws.
- •Multiple probability modes: Check exact, cumulative, tail, and range probabilities without manual summation.
- •Card probability support: Use it as a hypergeometric calculator for card probability, poker hands, and similar draw problems.
- •Built-in sanity checks: See the valid k range so impossible inputs are easier to catch.
- •Distribution context: Review mean, variance, and standard deviation alongside your selected probability.
For the factorial arithmetic behind combinations, use our Factorial Calculator to inspect n! values and edge cases.
Factors That Affect Your Results
Sampling Without Replacement
Every draw changes what remains in the population, so later draws depend on earlier draws. This drives the sampling without replacement probability formula.
Success Share K/N
A larger success share raises the expected number of successes and usually increases the probability of higher k values.
Sample Size n
Larger samples can make extreme outcomes more or less likely depending on how many successes and failures exist.
Valid Support Range
The possible k values may not start at 0 or end at n when the population has too few successes or failures.
According to OpenStax via Statistics LibreTexts, a hypergeometric experiment samples without replacement and the random variable counts items from the group of interest.
To compare with a continuous bell-curve model, explore our Normal Distribution Calculator to calculate z-scores and tail areas.
Frequently Asked Questions (FAQ)
Q: What is the hypergeometric distribution formula?
A: The hypergeometric distribution formula is P(X = k) = [C(K,k) x C(N-K,n-k)] / C(N,n). N is the population, K is the number of successes, n is the sample size, and k is the target successes.
Q: When should I use a hypergeometric calculator?
A: Use a hypergeometric calculator when you sample from a fixed population without replacement. Common examples include card hands, defective-item inspection, committee selection, inventory audits, and classroom probability problems where each draw changes what remains.
Q: What is the difference between hypergeometric and binomial distribution?
A: Hypergeometric probability is for draws without replacement, so trial probabilities change. Binomial probability is for independent trials with the same success chance each time. If the item is returned or the population is effectively unchanged, binomial may fit better.
Q: How do you calculate at least k successes?
A: To calculate at least k successes, add the hypergeometric probabilities from k through the maximum possible successes. This calculator does that summation automatically and also shows exact, at-most, fewer-than, greater-than, and between results.
Q: What do N, K, n, and k mean in a hypergeometric distribution?
A: N is the total population size, K is how many population items are successes, n is the number drawn, and k is the success count you want to test. All four values must be whole numbers.
Q: Can the hypergeometric distribution be used for card probability?
A: Yes. Card hands are a classic hypergeometric use case because cards are drawn without replacement. For example, N can be 52 cards, K can be 4 aces, n can be a 5-card hand, and k can be the number of aces.