Use our Hypergeometric Distribution Calculator to determine the probability of achieving a specific number of successes when items are randomly drawn from a finite population without replacement.

Hypergeometric Distribution Calculator

Enter the parameters for the hypergeometric distribution to calculate PDF, CDF, expected value, variance, and to generate a random sample.

* \(N\): population size, \(K\): number of successes in population, \(n\): sample size, \(k\): observed successes.

Results

PDF at \(k\):

CDF at \(k\):

Expected Value:

Variance:

Random Sample:

Hypergeometric Distribution: \( P(X=k)=\frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}} \), Expected Value = \( \frac{nK}{N} \), Variance = \( \frac{nK(N-K)(N-n)}{N^2(N-1)} \).

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Hypergeometric Distribution Calculator (In-Depth Explanation)

Hypergeometric Distribution Calculator (In-Depth Explanation)

The Hypergeometric Distribution models the probability of drawing a specific number of successes from a finite population without replacement. Unlike the binomial distribution, which assumes independent trials, the hypergeometric distribution applies when each draw affects the probabilities of subsequent draws. This guide explains the key concepts, the probability mass function, and provides a step-by-step process for calculating hypergeometric probabilities.

Table of Contents

  1. Overview of the Hypergeometric Distribution
  2. Key Concepts
  3. Distribution Function
  4. Step-by-Step Calculation Process
  5. Practical Examples
  6. Common Applications
  7. Conclusion

1. Overview of the Hypergeometric Distribution

The hypergeometric distribution describes the probability of obtaining exactly \( k \) successes in a sample of size \( n \) drawn without replacement from a finite population of size \( N \) that contains exactly \( K \) successes. It is commonly used in scenarios where sampling is done without replacement, such as quality control, lotteries, and ecological studies.

2. Key Concepts

To effectively use the Hypergeometric Distribution Calculator, consider these key parameters:

  • Population Size (\( N \)): The total number of items in the population.
  • Number of Successes in Population (\( K \)): The number of items classified as successes in the population.
  • Sample Size (\( n \)): The number of items drawn from the population.
  • Number of Successes in Sample (\( k \)): The random variable representing the count of successes in the drawn sample.
  • Sampling Without Replacement: Each draw affects the composition of the remaining population.

3. Distribution Function

The probability mass function (PMF) of the hypergeometric distribution is given by:

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

This formula calculates the probability of drawing exactly \( k \) successes in a sample of \( n \) items from a population of \( N \) items that contains \( K \) successes.

4. Step-by-Step Calculation Process

  1. Define the Parameters:

    Identify the total population size \( N \), the number of successes in the population \( K \), the sample size \( n \), and the number of successes \( k \) you wish to observe in the sample.

  2. Calculate the Binomial Coefficients:

    Compute the values for \( \binom{K}{k} \), \( \binom{N-K}{n-k} \), and \( \binom{N}{n} \). These coefficients represent the number of ways to choose successes and failures.

  3. Substitute into the Formula:

    Insert the computed coefficients into the PMF:

    \( P(X = k) = \frac{\binom{K}{k}\binom{N-K}{n-k}}{\binom{N}{n}} \)
  4. Compute the Probability:

    Evaluate the expression to obtain the probability of exactly \( k \) successes.

5. Practical Examples

Example: Quality Control in a Production Batch

Scenario: A production batch contains 100 items (\( N = 100 \)) with 10 defective items (\( K = 10 \)). A quality control inspector selects 15 items (\( n = 15 \)) at random. What is the probability that exactly 3 of the selected items are defective (\( k = 3 \))?

Step 1: Identify the parameters: \( N = 100 \), \( K = 10 \), \( n = 15 \), and \( k = 3 \).

Step 2: Compute the necessary binomial coefficients: \( \binom{10}{3} \), \( \binom{90}{12} \), and \( \binom{100}{15} \).

Step 3: Substitute the coefficients into the PMF:

\( P(X = 3) = \frac{\binom{10}{3}\binom{90}{12}}{\binom{100}{15}} \)

Step 4: Perform the calculation to determine the probability.

6. Common Applications

  • Quality Control: Estimating the number of defective items in a lot.
  • Ecology: Assessing the distribution of species in a habitat.
  • Lotteries and Games: Calculating odds when drawing without replacement.
  • Statistical Sampling: Analyzing results in finite populations.

7. Conclusion

The Hypergeometric Distribution Calculator offers a systematic approach for evaluating probabilities in scenarios involving sampling without replacement. By applying the probability mass function:

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

users can accurately determine the likelihood of observing a specific number of successes in a sample drawn from a finite population. This tool is essential for applications in quality control, ecological studies, and other fields where sampling without replacement is critical.

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