Hypergeometric Distribution Calculator

Enter your population size (N), number of successes in population (K), sample size (n), and desired successes (k) to calculate exact and cumulative hypergeometric probabilities.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter Population Size (N)
Input the total number of items in the population from which you are sampling.
2
Specify Successes in Population (K)
Provide the number of 'successful' items present within the total population.
3
Input Sample Size (n)
Enter the number of items you are drawing from the population without replacement.
4
Specify Desired Successes (k)
Indicate the exact number of successes you want to find the probability for within your sample.
5
Review Probability and Statistical Measures
The calculator will display the exact, cumulative, and upper-tail probabilities, along with the mean, variance, and standard deviation.

Example Calculation

A quality control manager is inspecting a batch of 50 items, 10 of which are known to be defective. They draw a sample of 5 items and want to know the probability of finding exactly 2 defective items.

Population Size

Successes in Population

Sample Size

Desired Successes

Results

0.209849

Tips

Distinguish from Binomial

Remember that hypergeometric distribution applies when sampling *without replacement* from a finite population, meaning each draw affects subsequent probabilities. The binomial distribution, conversely, applies to sampling *with replacement* or from an infinite population.

Verify Input Constraints

Always ensure that `k ≤ n` (desired successes cannot exceed sample size) and `k ≤ K` (desired successes cannot exceed total successes in population). Also, `n ≤ N` (sample size cannot exceed population size) for valid results.

Interpret Cumulative Probability

The cumulative probability P(X ≤ k) is often more useful than exact probability in real-world scenarios, as it tells you the chance of getting 'at most k' successes. For example, a P(X ≤ 2) of 0.95 means there's a 95% chance of finding 2 or fewer successes.

Calculating Probabilities for Sampling Without Replacement: The Hypergeometric Distribution

The Hypergeometric Distribution Calculator is an invaluable tool for determining probabilities when sampling from a finite population without replacement. It provides exact, cumulative, and upper-tail probabilities, along with essential statistical measures like mean, variance, and standard deviation. For instance, if you draw 5 items from a population of 50 containing 10 successes, the probability of finding exactly 2 successes is approximately 0.209849. This calculation is crucial in fields ranging from quality control to genetics in 2025.

Probabilistic Benchmarks in Statistical Analysis

Probabilities derived from distributions like the hypergeometric are fundamental benchmarks in various statistical analyses. For example, in hypothesis testing, a p-value of less than 0.05 (5%) is a widely accepted benchmark for statistical significance, indicating that an observed result is unlikely to have occurred by random chance. In quality control, acceptance sampling plans often use a 0.01% (1 in 10,000) or even 0.001% chance of accepting a batch with a critical defect rate as a benchmark for product reliability. Furthermore, in fields like genetics, the probability of inheriting specific traits or diseases often falls into very small percentages, requiring precise calculations to inform risk assessments and counseling. These benchmarks provide a standardized way to interpret statistical outcomes and make informed decisions.

The Hypergeometric Distribution Formula Explained

The probability mass function (PMF) for the hypergeometric distribution is given by the formula:

P(X = k) = [C(K, k) × C(N - K, n - k)] / C(N, n)

Where:

P(X = k) is the probability of observing exactly k successes in the sample.
N is the population size.
K is the number of successes in the population.
n is the sample size.
k is the number of desired successes in the sample.
C(a, b) denotes the number of combinations of choosing b items from a, calculated as a! / (b! × (a-b)!).

💡 For analyzing growth over time, especially in biological or economic contexts, our Population Growth Percentage Calculator offers a different probabilistic perspective.

Calculating Probability for a Quality Control Sample

Let's calculate the exact probability of finding 2 defective items in a sample of 5, drawn from a batch of 50 items known to contain 10 defectives.

Population Size (N): 50
Successes in Population (K): 10 (defective items)
Sample Size (n): 5
Desired Successes (k): 2

We use the combinations formula C(a, b) = a! / (b! * (a-b)!):

C(K, k) = C(10, 2) = 10! / (2! × 8!) = (10 × 9) / (2 × 1) = 45
C(N - K, n - k) = C(50 - 10, 5 - 2) = C(40, 3) = 40! / (3! × 37!) = (40 × 39 × 38) / (3 × 2 × 1) = 9880
C(N, n) = C(50, 5) = 50! / (5! × 45!) = (50 × 49 × 48 × 47 × 46) / (5 × 4 × 3 × 2 × 1) = 2,118,760

Now, substitute these into the hypergeometric formula: P(X = 2) = (45 × 9880) / 2,118,760 = 444,600 / 2,118,760 ≈ 0.209849

The exact probability of finding 2 defective items in the sample is approximately 0.209849, or about 21.0%.

💡 For analyzing sports data and other performance metrics, our Possession Percentage Calculator can provide insights into event distribution, similar to how this tool analyzes success rates.

Hypergeometric Distribution in Quality Control and Sampling

The hypergeometric distribution is a cornerstone in quality control and various sampling methodologies where items are drawn without replacement. For instance, a common application is acceptance sampling, where a manufacturer inspects a small sample from a larger batch of products to decide whether to accept or reject the entire batch. If a batch of 1,000 components is known to contain 20 defectives, and an inspector draws a sample of 50, the hypergeometric distribution can calculate the probability of finding exactly 0, 1, or 2 defectives in that sample. This helps set appropriate sampling plans and defect thresholds. Similarly, in ecological studies, it can model the probability of capturing a specific number of tagged animals in a subsequent trapping event from a finite population, providing insights into population dynamics and estimation in 2025.

Probabilistic Benchmarks in Statistical Analysis

Frequently Asked Questions

What is the hypergeometric distribution?

The hypergeometric distribution is a discrete probability distribution that describes the probability of drawing a specific number of 'successes' in a sample, drawn *without replacement*, from a finite population. Unlike the binomial distribution, each item drawn affects the probability of subsequent draws, making it suitable for scenarios like quality control inspections where items are not returned to the batch after sampling.

When should I use the hypergeometric distribution?

You should use the hypergeometric distribution when you are sampling from a finite population, and the sampling is done *without replacement*. This means that once an item is selected for the sample, it cannot be selected again. Common applications include quality control, card games (like poker), and biological sampling, where the population size is fixed and known.

What are the key parameters of the hypergeometric distribution?

The hypergeometric distribution is defined by four key parameters: Population Size (N), which is the total number of items; Successes in Population (K), the total number of items with the desired characteristic; Sample Size (n), the number of items drawn; and Desired Successes (k), the specific number of successes you are looking for in your sample. These parameters collectively determine the probability outcomes.

How does 'sampling without replacement' affect probabilities?

Sampling without replacement means that once an item is selected from the population, it is not put back. This action changes the composition of the remaining population, altering the probabilities for subsequent draws. For example, if you draw a red ball from a bag and don't replace it, the probability of drawing another red ball changes because there are fewer red balls and fewer total balls left in the bag.