About this calculator
The Hypergeometric Distribution Calculator is used to calculate probabilities in sampling without replacement. A typical question is: There are N objects in the population, K of which are successful types. If n objects are drawn from them without replacement, what is the probability of exactly k successful types being drawn.
The probability formula of hypergeometric distribution is P(X=k)=C(K,k)C(N-K,n-k)/C(N,n). It differs from the binomial distribution in whether sampling is done with replacement: the binomial distribution assumes a constant probability of success for each trial, whereas in the hypergeometric distribution each draw changes the remaining population structure.
This distribution is commonly used in quality inspection, lottery probabilities, inventory sampling, poker problems, and biostatistics. The calculator can help you quickly derive probabilities, understand the meaning of parameters, and avoid hand calculation errors of combinatorial numbers.
What it calculates
The hypergeometric distribution calculator finds the probability of getting a chosen number of successes when sampling without replacement from a finite population.
Formula
P(X = k) = C(K, k) * C(N - K, n - k) / C(N, n). N is population size, K is successes in the population, n is sample size, and k is successes drawn.
Inputs
- N: population size.
- K: number of success states in the population.
- n: number of draws.
- k: desired number of successes.
Example
| Scenario | Parameters | Question |
|---|---|---|
| Cards | N=52, K=4, n=5 | Aces in a 5-card hand |
| Quality check | N=100, K=8, n=10 | Defective items in 10 samples |
| Lottery | N=50, K=5, n=3 | Winning items in 3 draws |
How to interpret the result
The result is the probability of exactly k successes without replacement. After each draw, the population changes, which is the key difference from a binomial model.
Common mistakes
- Hypergeometric distribution is for sampling without replacement.
- k cannot exceed K or n.
- n cannot exceed population size N.
- Do not mix it with binomial distribution for independent repeated trials.
How to use
Enter the population number N, the number of successful objects K, the sampling number n, and the number of successes you want to calculate k. After clicking "Calculate", the tool will give the probability based on the hypergeometric distribution formula.
For example, there are 5 defective products in a batch of 50 products. If 10 products are randomly inspected, find the probability of picking out exactly 2 defective products. At this time, N=50, K=5, n=10, k=2, just substitute it into the formula.
When inputting, ensure that 0≤K≤N, 0≤n≤N, and k cannot exceed K or n, nor be less than n-(N-K). Otherwise the event cannot occur, the probability is 0, or the input is invalid.
Main features
Supports sampling probability calculation without replacement.
Explain the meaning of N, K, n, k using the combinatorial number formula for exactly k successes, range probability, and expected variance learning.
Ideal for quality control, lottery analysis, poker and statistics courses to reduce calculation errors in large combinations.
Use cases
In quality inspection, hypergeometric distribution can be used to estimate the probability of finding defective products in sampling samples and help formulate sampling plans.
In probability courses, playing cards, ball box sampling and lottery without replacement are all classic question types of hypergeometric distribution.
In biostatistics and survey research, hypergeometric models can be more accurate than binomial models when samples are drawn from finite populations and without replacement.