Hypergeometric Distribution Probability Calculator

Probability

Population size

Number of successes in population

Sample size

Number of successes in sample (x)

Minimum successes (x₁)

Maximum successes (x₂)

Results

P(X = x)

Cumulative probability: P(X < x)

Cumulative probability: P(X ≤ x)

Cumulative probability: P(X > x)

Cumulative probability: P(X ≥ x)

P(x₁ ≤ X ≤ x₂)

P(X < x₁) + P(X > x₂)

Mean

Standard deviation

Grasping the concept of probabilities can be challenging, particularly in situations involving sampling without replacement. Our Hypergeometric Distribution Probability Calculator simplifies this task, enabling you to calculate probabilities quickly and accurately.

What Is the Hypergeometric Distribution Probability?

The hypergeometric distribution probability is used when you want to determine the likelihood of getting a specific number of successes without replacement from a finite population.

It applies when:

The population is finite
Each item can be classified as a success or a failure
Samples are drawn without replacement
The order of selection does not matter

The hypergeometric distribution is different from the binomial distribution because the binomial assumes replacement, while the hypergeometric does not.

Hypergeometric Distribution Formula

The probability of obtaining x successes in n draws from a population of size N containing K successes is:

Formula:

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

Where:

N = Population size

K = Total number of successes in the population

n = Sample size

x = Number of successes in the sample

(\binom{a}{b}) = Combination function (a choose b)

Hypergeometric Distribution Probability (Range: Minimum to Maximum Successes)

To calculate the probability of getting between a minimum number of successes (x_min) and a maximum number of successes (x_max) in a hypergeometric setting, you sum the individual probabilities for each valid value of x.

Formula:

P (x_{\min} \leq X \leq x_{\max}) = \sum_{x = x_{\min}}^{x_{\max}} \frac{(\binom{K}{x}) \cdot (\binom{N - K}{n - x})}{(\binom{N}{n})}

Where:

N = Population size

K = Number of successes in the population

n = Sample size

x = Number of successes in the sample

x_min, x_max = Minimum and maximum success range

(\binom{a}{b}) = Combination function (a choose b)

Mean and Standard Deviation of the Hypergeometric Distribution

Formula for Mean:

μ = n \frac{k}{N}

Formula for Standard Deviation:

σ^{2} = n \frac{k}{N} \frac{N - k}{N} \frac{N - n}{N - 1}

Where:

n = the number of occurrences;

k = is the number of successes;

N = is the population size.

How to Use Our Hypergeometric Distribution Probability Calculator

Select the Probability type for which the value has to be determined, i.e., for a specific value of x or for a range.
Enter the values in the respective fields.
The calculator computes P(X=x), P(X < x), P(X ≤ x), P(X > x), P(X ≥ x), Mean, and Standard Deviation.