Hypergeometric Distribution Calculator

Population Size

N

Successes in Pop

K

Sample Size

n

Probability Type:

Successes in Sample

x

Upper Bound

x_{2}

Output Type: Chart Table

Decimal Places: 2 3 5 8

Clear

Introduction

The hypergeometric distribution models the probability of observing $x$ successes in a sample of size $n$ drawn without replacement from a finite population of size $N$ containing $K$ successes. Analysing this structure captures how the likelihood of success changes from trial to trial and supports the study of dependent sampling processes within discrete probability theory.

What this calculator does

The tool processes four primary inputs: population size, total successes within that population, sample size, and the desired number of successes in the sample. It generates the probability for specific points or ranges, including cumulative probabilities. Additionally, it provides the statistical mean and variance for the given distribution, alongside a visual chart or detailed data table for comprehensive analysis.

Formula used

The probability mass function relies on combinations to account for sampling without replacement. Here, $N$ is the population size, $K$ is the number of successes in the population, $n$ is the sample size, and $x$ is the number of observed successes.

P (X = x) = \frac{(\binom{K}{x}) (\binom{N - K}{n - x})}{(\binom{N}{n})}

E [X] = n \frac{K}{N}

How to use this calculator

1. Enter the total population size $N$ and the number of successes $K$ within that population.
2. Input the sample size $n$ and select the desired probability type, such as equal to, less than, or between.
3. Specify the number of successes $x$ to be measured within the sample.
4. Execute the calculation to view the probability, mean, variance, and the selected visual output.

Example calculation

Scenario: A social research study examines a small group of 52 individuals, 4 of whom possess a specific rare trait, to find the probability of selecting 1 person with that trait in a sample of 5.

Inputs: Population size $N = 52$ , successes in population $K = 4$ , sample size $n = 5$ , and successes in sample $x = 1$ .

Working:

Step 1: $P (X = x) = \frac{(\binom{K}{x}) (\binom{N - K}{n - x})}{(\binom{N}{n})}$

Step 2: $P (X = 1) = \frac{(\binom{4}{1}) (\binom{52 - 4}{5 - 1})}{(\binom{52}{5})}$

Step 3: $P (X = 1) = \frac{(4) (194580)}{2598960}$

Step 4: $P (X = 1) = 0.29947364$

Result: 0.30 (rounded to 2 decimal places).

Interpretation: There is a 30% chance that exactly one person in the sample of five will possess the specific trait.

Summary: The calculation provides a precise probability for discrete outcomes in finite populations.

Understanding the result

The resulting probability indicates the likelihood of observing a specific count of successes. A high probability suggests the outcome is expected given the population density of successes, while the mean and variance reveal the central tendency and spread of the distribution across all possible sample outcomes.

Assumptions and limitations

The calculation assumes the population is finite and that sampling is conducted without replacement, meaning each trial is dependent on the previous one. It also requires all inputs to be non-negative integers where the sample size does not exceed the population size.

Common mistakes to avoid

Users often confuse this distribution with the binomial distribution, which assumes sampling with replacement and independent trials. Another error is entering a sample size $n$ or success count $x$ that is larger than the total population $N$ or population successes $K$ , leading to invalid results.

Sensitivity and robustness

The output is highly sensitive to the ratio of successes in the population. Small changes in the population success count $K$ or the sample size $n$ can significantly shift the mean and individual probabilities. The calculation remains stable within the permitted educational range of 100,000 for input parameters.

Troubleshooting

If the result returns zero, ensure that the successes requested $x$ are mathematically possible; for instance, $x$ cannot exceed $K$ or $n$ . Error messages will appear if inputs are non-integers or if the requested bounds for a "between" probability are logically inverted.

Frequently asked questions

How does this differ from Binomial distribution?

The hypergeometric distribution is used for sampling without replacement from a finite population, whereas the binomial distribution assumes independent trials with replacement.

What is the maximum input value?

The calculator allows for population and sample parameters up to 100,000 for educational purposes.

What is the meaning of the variance in this context?

The variance measures how much the number of successes in the sample is expected to deviate from the mean value across many repeated sampling instances.

Where this calculation is used

In academic environments, this distribution is a fundamental concept in probability theory and modelling. It is frequently applied in environmental science for population studies, such as estimating the number of members of a species in a specific area based on tag-and-recapture methods. Social researchers use it to analyse demographic traits within small, finite groups where sampling one individual affects the likelihood of selecting another. It also serves as a critical component in descriptive statistics for understanding discrete variables in finite systems.

Results are based on standard mathematical and statistical methods and may involve rounding or approximation. If precise accuracy is required, please verify results independently. See full disclaimer.