Hypergeometric Distribution Calculator

- May 09, 2025

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Calculate probabilities for sampling without replacement from a finite population of successes and failures. Useful for quality control, card games, lottery analysis, and population sampling.

Population Parameters

Population Size (N):

Successes in Population (K):

Sample Parameters

Sample Size (n):

Successes in Sample (k):

Calculation Type

Probability Type:

Display Options

Decimal Places:

Show calculation steps

Show probability distribution

Hypergeometric Distribution Results

Probability (X = 2)

0.0000

Expected Value (Mean)

0.0000

Variance

0.0000

Standard Deviation

0.0000

Probability Distribution

Successes (k)	Probability P(X = k)	Cumulative P(X ≤ k)

Calculation Steps

Formula Used

The hypergeometric probability mass function for obtaining exactly k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes:

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

About Hypergeometric Distribution

The hypergeometric distribution describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes.

Key Properties

Mean: μ = n × (K/N)
Variance: σ² = n × (K/N) × (N-K)/N × (N-n)/(N-1)
Sampling: Without replacement from a finite population
Difference from Binomial: Probabilities change after each draw

Applications

Quality Control: Probability of finding defects in a sample from a batch
Card Games: Calculating probabilities in poker, bridge, etc.
Lottery Analysis: Probability of matching a certain number of winning numbers
Audit Sampling: Determining likelihood of finding errors in financial records
Population Studies: Analyzing traits in samples from a known population

What is the Hypergeometric Distribution?

The hypergeometric distribution is a way to understand probabilities when drawing items from a limited group. Imagine you have a jar filled with red and blue marbles. If you want to know the chance of picking a certain number of red marbles without putting any back, the hypergeometric distribution can help. This method is particularly useful when the population is finite, which means there’s a set number of items to choose from.

How Does the Hypergeometric Distribution Calculator Work?

The Hypergeometric Distribution Calculator allows you to calculate probabilities based on known values. You’ll need to enter the total number of items, the number of successful items, the sample size, and how many successful outcomes you’re looking for. With this information, the calculator computes the probabilities for various scenarios, helping you make informed decisions.

Key Features of the Calculator

Some key aspects of the Hypergeometric Distribution Calculator include:

Population Parameters: Input the total size of your population and the number of successes in that group.
Sample Parameters: Define how many items you're drawing and the successes you wish to find.
Calculation Types: Choose whether you want an exact probability, a cumulative probability, or a range of probabilities.

These features make it user-friendly and adaptable to various needs.

Applications of the Hypergeometric Distribution

This calculator is useful in many fields. Here are a few:

Quality Control: Determine the likelihood of defects in a sample from a larger batch.
Card Games: Assess probabilities in games like poker and bridge.
Lottery Analysis: Calculate the chances of winning by matching a specific number of items.
Survey Analysis: Find out how representative a sample is of a larger population.

These applications highlight the versatility of the hypergeometric distribution in real-world situations.

Understanding the Results

When you use the Hypergeometric Distribution Calculator, you’ll receive results that include the probability of achieving the desired number of successes. You’ll also see the expected value, variance, and standard deviation of your sample. These results give a clearer picture of your data and help interpret the findings accurately.

How to Use the Calculator Effectively

To make the most of the Hypergeometric Distribution Calculator:

Input Accurate Values: Ensure that the numbers you enter reflect your actual situation.
Select the Right Calculation Type: Depending on what you need, choose the exact or cumulative probabilities.
Review the Distribution Chart: Visual representations can help you understand the probability distribution better.

These steps will enhance your experience and improve the accuracy of your results.

The Formula Behind the Calculator

The calculator uses a specific formula to calculate probabilities:

[ P(X = k) = \frac{{(K \text{ choose } k) \times (N-K \text{ choose } n-k)}}{{(N \text{ choose } n)}} ]

This equation takes into account the total population, the number of successes, and the sample size. Understanding this formula can deepen your grasp of how probabilities are calculated and what they mean for your scenario.

Why Choose the Hypergeometric Distribution Calculator?

Choosing the Hypergeometric Distribution Calculator means you’re opting for a reliable tool that simplifies complex calculations. Whether you’re in quality control, examining card Game odds, or working on a research project, this calculator provides clarity and precision. It’s a straightforward way to handle probabilities without getting lost in complicated math, making it an excellent resource for anyone needing to analyse finite populations.

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