Hypergeometric Distribution Probability Calculator

The hypergeometric distribution calculates probabilities of successes in a sample drawn without replacement from a finite population. This calculator allows you to compute exact, less than, greater than, and cumulative probabilities, and also visualizes the hypergeometric distribution.

Population Size (N):

Number of Successes in Population (K):

Sample Size (n):

Number of Successes in Sample (x):

P(X = ):

P(X < ):

P(X ≤ ):

P(X > ):

P(X ≥ ):

Understanding the Hypergeometric Distribution

The Hypergeometric distribution is a discrete probability distribution that describes the probability of drawing a specific number of successes from a sample drawn without replacement from a finite population. It is commonly used in scenarios where sampling is done without replacement, such as quality control testing or card games.

Key Components of the Hypergeometric Distribution

Population Size (N): The total number of items in the population.
Number of Successes in Population (K): The total number of items in the population that are classified as successes.
Sample Size (n): The number of items drawn from the population (without replacement).
Number of Successes in Sample (x): The number of items in the sample that are classified as successes.

Hypergeometric Distribution Formula

The probability mass function (PMF) for the Hypergeometric distribution is given by the formula:

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

Here, \( \binom{K}{x} \) represents the number of ways to choose \( x \) successes from \( K \) successes in the population, and \( \binom{N-K}{n-x} \) represents the number of ways to choose \( n-x \) failures from the remaining \( N-K \) items in the population. Finally, \( \binom{N}{n} \) is the total number of ways to choose \( n \) items from the population.

Step-by-Step Example: Finding Hypergeometric Probability

Suppose we want to calculate the probability of drawing exactly 5 successes from a population of 50 items, where 20 items are classified as successes. We draw a sample of 10 items without replacement.

Step 1: Identify the Key Parameters

In this case:

\( N = 50 \): The total population size.
\( K = 20 \): The number of successes in the population.
\( n = 10 \): The sample size.
\( x = 5 \): The number of successes in the sample.

Step 2: Apply the Hypergeometric Formula

Using the formula:

\[ P(X = 5) = \frac{\binom{20}{5} \binom{30}{5}}{\binom{50}{10}} \]

First, calculate the combinations:

\( \binom{20}{5} = 15504 \)
\( \binom{30}{5} = 142506 \)
\( \binom{50}{10} = 10272278170 \)

Substituting these values into the formula:

\[ P(X = 5) = \frac{15504 \times 142506}{10272278170} \approx 0.21509 \]

Therefore, the probability of drawing exactly 5 successes from this sample is approximately 0.21509, or 21.51%.

Step 3: Additional Probabilities

In addition to finding the probability of drawing exactly 5 successes, we can calculate other probabilities using the cumulative distribution:

P(X = 5): 0.21509
P(X < 5): 0.64503
P(X ≤ 5): 0.86011
P(X > 5): 0.13989
P(X ≥ 5): 0.35497

Conditions for Using the Hypergeometric Distribution

Sampling without Replacement: The hypergeometric distribution applies when items are drawn without replacement, meaning that each draw changes the composition of the population.
Finite Population: The population size \( N \) must be finite.