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
Hypergeometric Distribution Calculator is based on the complete Chinese reference article for this calculator. It explains what the tool calculates, when to use it, and how the result relates to the underlying formula.
Formula
Use the formula shown by Hypergeometric Distribution Calculator together with the values entered in the calculator. Keep units consistent and check any restrictions before interpreting the answer.
Inputs
Enter the required values for Hypergeometric Distribution Calculator. Use numeric inputs where requested, keep variable names consistent, and review the selected unit or calculation mode before calculating.
- Required numeric values.
- Relevant units or variable names.
- Calculation mode or target value when available.
Example
A typical example uses simple values so you can compare the input, formula, and output. This helps verify that the calculator is being used correctly.
| Step | What to check | Purpose |
|---|---|---|
| 1 | Enter sample values | Confirm how Hypergeometric Distribution Calculator reads inputs |
| 2 | Review the formula | Understand the calculation method |
| 3 | Compare the result | Use the answer correctly |
How to interpret the result
The result should be read together with the formula, input values, and any displayed calculation steps. If the calculator shows multiple values, compare each label before using the answer.
Common mistakes
Most mistakes come from missing units, entering values in the wrong field, or ignoring formula restrictions. Recheck the inputs if the result looks unexpected.
- Check units and signs.
- Do not leave required inputs blank.
- Confirm that the formula conditions are satisfied.
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.