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.

**P(X = ):**

**P(X < ):**

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## 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:

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:

First, calculate the combinations:

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

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.

### Further Reading

Suf is a senior advisor in data science with deep expertise in Natural Language Processing, Complex Networks, and Anomaly Detection. Formerly a postdoctoral research fellow, he applied advanced physics techniques to tackle real-world, data-heavy industry challenges. Before that, he was a particle physicist at the ATLAS Experiment of the Large Hadron Collider. Now, he’s focused on bringing more fun and curiosity to the world of science and research online.