Hypergeometric Distribution Calculator

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Parameters

Population size (N):

Success states in population (K):

Number of draws (n):

Calculation Type:

Number of successes (x):

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Hypergeometric Distribution: Definition, Formula, and Examples

Hypergeometric Distribution

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

Formula:

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

Where:

$N$ is the population size
$K$ is the number of success states in the population
$n$ is the number of draws
$k$ is the number of successes in the sample
$\binom{n}{k}$ represents the binomial coefficient

Example: Imagine a box containing 20 marbles, of which 8 are blue and 12 are red. If you draw 5 marbles without replacement, what is the probability of getting exactly 3 blue marbles?

In this case:

$N = 20$ (total marbles)
$K = 8$ (blue marbles - success states)
$n = 5$ (draws)
$k = 3$ (desired blue marbles)

Plugging these values into the formula:

P(X = 3) = \frac{\binom{8}{3}\binom{12}{2}}{\binom{20}{5}} = \frac{56 \cdot 66}{15504} \approx 0.238

Therefore, the probability of drawing exactly 3 blue marbles is about 23.8%.

Properties

Mean: $E(X) = n\frac{K}{N}$
Variance: $\text{Var}(X) = n\frac{K}{N}\frac{N-K}{N}\frac{N-n}{N-1}$
Support: $\text{max}(0, n-(N-K)) \leq k \leq \text{min}(n,K)$

Hypergeometric Distribution Calculator

Calculator

Parameters

Distribution Chart

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Hypergeometric Distribution: Definition, Formula, and Examples

Hypergeometric Distribution

Properties

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