Hypergeometric Distribution Calculator
Our free exponents & logarithms calculator solves hypergeometric distribution problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Hypergeometric Distribution Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: P(X=k) = C(K,k) * C(N-K, n-k) / C(N,n)
Worked example — P(X=2) = 0.2743 or 27.43%
Formula
P(X=k) = C(K,k) * C(N-K, n-k) / C(N,n)
Where N is the population size, K is the number of success states in the population, n is the number of draws (sample size), and k is the desired number of observed successes. C(a,b) is the binomial coefficient 'a choose b'.
Worked Examples
Example 1: Drawing Hearts from a Deck
Problem:What is the probability of drawing exactly 2 hearts in a 5-card hand from a standard 52-card deck (13 hearts)?
Solution:N = 52 (population), K = 13 (hearts), n = 5 (draw), k = 2 (desired hearts) P(X=2) = C(13,2) * C(39,3) / C(52,5) C(13,2) = 78 C(39,3) = 9,139 C(52,5) = 2,598,960 P(X=2) = 78 * 9,139 / 2,598,960 = 712,842 / 2,598,960
Result:P(X=2) = 0.2743 or 27.43%
Example 2: Quality Control Inspection
Problem:A lot of 200 items contains 15 defective items. If 10 items are inspected, what is the probability of finding exactly 1 defective?
Solution:N = 200, K = 15, n = 10, k = 1 P(X=1) = C(15,1) * C(185,9) / C(200,10) Mean = 10 * 15/200 = 0.75 P(X=1) = 15 * C(185,9) / C(200,10)
Result:P(X=1) = 0.3670 or 36.70%
Frequently Asked Questions
What is the hypergeometric distribution?
The hypergeometric distribution models the probability of drawing a specific number of successes from a finite population without replacement. Unlike the binomial distribution which assumes replacement (or infinite population), the hypergeometric distribution accounts for the changing probability as items are drawn. A classic example is drawing cards from a deck: what is the probability of getting exactly 2 hearts in a 5-card hand from a standard 52-card deck? The distribution is defined by three parameters: the population size N, the number of success states K in the population, and the number of draws n. Each draw changes the composition of the remaining population.
How is the hypergeometric distribution different from the binomial distribution?
The key difference is sampling with versus without replacement. The binomial distribution assumes each trial is independent with a constant probability of success, which applies when sampling with replacement or from an effectively infinite population. The hypergeometric distribution accounts for the fact that each draw changes the remaining population composition. For example, after drawing a heart from a deck, the probability of the next card being a heart changes from 13/52 to 12/51. When the population is very large relative to the sample size, the hypergeometric distribution approximates the binomial distribution because removing one item barely changes the probabilities.
What is the formula for the hypergeometric probability?
The probability mass function is P(X = k) = C(K,k) * C(N-K, n-k) / C(N,n), where C(a,b) is the binomial coefficient (a choose b). Here N is the total population, K is the number of success items, n is the sample size, and k is the desired number of successes. The numerator counts the favorable outcomes: C(K,k) ways to choose k successes from K success items, times C(N-K, n-k) ways to choose the remaining n-k items from the N-K non-success items. The denominator C(N,n) counts all possible ways to draw n items from N. This ratio gives the exact probability.
What are common applications of the hypergeometric distribution?
The hypergeometric distribution appears in quality control when inspecting a batch of products without replacement, such as testing 10 items from a batch of 100 to check for defects. It is used in ecology for capture-recapture methods to estimate animal populations. In card games, it calculates the probability of specific hands. In genetics, it models the likelihood of observing a certain number of genes of interest in a random sample. Statistical tests like Fisher's exact test use the hypergeometric distribution for analyzing contingency tables, especially with small sample sizes where chi-squared approximations are unreliable.
What are the mean and variance of the hypergeometric distribution?
The mean (expected value) of the hypergeometric distribution is E(X) = nK/N, which is intuitive: if 25% of the population are successes and you draw 10 items, you expect 2.5 successes on average. The variance is Var(X) = n*K*(N-K)*(N-n) / (N^2*(N-1)). The factor (N-n)/(N-1) is called the finite population correction factor, and it makes the variance smaller than the corresponding binomial variance. As the population grows much larger than the sample, this correction factor approaches 1 and the variance approaches the binomial variance of n*p*(1-p) where p = K/N.
How do you calculate cumulative hypergeometric probabilities?
Cumulative probabilities are computed by summing individual probabilities. P(X <= k) sums P(X = x) for all valid x from the minimum possible successes to k. The minimum is max(0, n+K-N), which accounts for cases where you must draw some successes because there are not enough non-successes to fill the sample. The maximum possible successes is min(K, n). For P(X >= k), compute 1 - P(X <= k-1). These cumulative values answer practical questions like: what is the probability of getting at least 3 defective items when inspecting 10 items from a lot where 20 out of 200 are defective?
When should I use the hypergeometric distribution instead of other distributions?
Use the hypergeometric distribution whenever you are sampling without replacement from a finite, well-defined population with two categories (success/failure). The critical factor is whether the sample size is a significant fraction of the population. A common rule of thumb is to use the hypergeometric when the sample is more than 5-10% of the population. Below that threshold, the binomial distribution provides a good approximation and is computationally simpler. For very large populations with moderate success probabilities, the Poisson distribution may also serve as an approximation. Always use the exact hypergeometric distribution for small populations.
What is Fisher's exact test and how does it use the hypergeometric distribution?
Fisher's exact test is a statistical test used to determine if there is a significant association between two categorical variables in a 2x2 contingency table. It calculates the exact probability of observing the given arrangement (or a more extreme one) under the null hypothesis of no association. The probability is computed using the hypergeometric distribution because the test conditions on the row and column totals being fixed, making it equivalent to sampling without replacement. Unlike the chi-squared test, Fisher's exact test is valid for any sample size and is particularly important when expected cell counts are small (below 5).
How does sample size affect hypergeometric probabilities?
Increasing the sample size generally makes the distribution less variable and more concentrated around the mean. With a larger sample, you are more likely to observe a proportion of successes close to the population proportion K/N. However, as the sample size approaches the population size, the distribution becomes degenerate because you are sampling nearly the entire population, leaving little randomness. For a fixed population and success count, larger samples also shift the range of possible outcomes. The minimum possible successes increases when n+K exceeds N, because you must draw some successes. This constraint becomes more pronounced as sample sizes grow.
Can the hypergeometric distribution be multivariate?
Yes, the multivariate hypergeometric distribution extends the basic version to more than two categories. Instead of just success/failure, you might have multiple types of items in the population. For example, drawing 5 cards and wanting to know the probability of getting 2 hearts, 1 spade, 1 diamond, and 1 club. The formula generalizes to a product of binomial coefficients in the numerator (one for each category) divided by the total combinations. This distribution is used in ecology for sampling species, in manufacturing for multi-defect-type quality control, and in survey sampling where respondents fall into multiple demographic categories.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
Related Calculators
🧮Poisson Distribution Calculator
Calculate poisson distribution with inputs, formulas, and instant results.
🧮Geometric Distribution Calculator
Calculate geometric distribution with inputs, formulas, and instant results.
🧮Normal Distribution Calculator
Calculate normal distribution with inputs, formulas, and instant results.
🧮Exponential Distribution Calculator
Calculate exponential distribution probabilities from rate parameter and time interval.
🧮Z-Score Calculator with Step-by-Step Solution
Calculate z-scores and find probabilities from the standard normal distribution table.
🧮Annulus Area Calculator
Calculate annulus area with inputs, formulas, and instant results.
🧮Area Calculator
Calculate area with inputs, formulas, and instant results.
🧮Area of a Rectangle Calculator
Calculate the area, perimeter, and diagonal of a rectangle. Find missing sides from known area. Convert between metric and imperial area units.