# Hypergeometric Distribution Calculator

Calculate exact and cumulative probabilities for sampling without replacement

> Canonical page: https://elysiatools.com/en/tools/hypergeometric-distribution-calculator

- **Category:** Math & Numbers

- **Keywords:** hypergeometric distribution, sampling without replacement, probability, statistics

## Overview

Calculate hypergeometric probabilities when drawing without replacement from a finite population with a known number of successes.

## Inputs

- **Population Size** (number)
- **Population Successes** (number)
- **Draws** (number)
- **Observed Successes** (number)
- **Probability Mode** (select)
- **Decimal Places** (number)

## When to use

- When calculating probabilities for card games or lotteries where drawn items are not returned to the pool.
- During quality control inspections to determine the likelihood of finding defective items in a sample batch.
- In ecological or demographic studies involving capture-recapture methods from a finite population.

## How it works

- Enter the total Population Size and the number of Population Successes (e.g., total items and total target items).
- Specify the number of Draws (sample size) and the target Observed Successes.
- Select the Probability Mode to calculate exact, at-most, or at-least probabilities.
- Adjust the decimal places to generate the precise probability result in JSON format.

## Use cases

- Quality assurance teams testing a random sample of manufactured goods for defects without returning tested items to the batch.
- Data analysts calculating the exact odds of drawing specific card combinations in poker or trading card games.
- Auditors reviewing a subset of financial records to estimate the probability of compliance errors in a finite ledger.

## Frequently asked questions

### What is the difference between hypergeometric and binomial distributions?

Hypergeometric distribution models sampling without replacement, meaning probabilities change after each draw. Binomial distribution assumes sampling with replacement where probabilities remain constant.

### What does 'Population Successes' mean?

It refers to the total number of items in the entire population that match the characteristic you are looking for, such as the total number of winning tickets or defective parts.

### Can I calculate cumulative probabilities?

Yes, use the Probability Mode dropdown to select 'At Most k Successes' or 'At Least k Successes' for cumulative results.

### What are the limits for population size and draws?

The calculator supports population sizes, successes, draws, and observed successes up to 10,000.

### Why must the observed successes be less than or equal to the draws?

You cannot draw more successful items than the total number of items you are drawing from the population.

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