Hypergeometric Distribution Calculator

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Success states in population (K):

Number of draws (n):

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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: $\text{max}(0, n-(N-K)) \leq k \leq \text{min}(n,K)$

Applications

1. Quality Control and Manufacturing

In manufacturing quality control, the hypergeometric distribution helps calculate the probability of finding defective items in a sample drawn from a batch. For example, when inspecting a batch of products by sampling without replacement, it can determine the likelihood of finding a specific number of defective items.

2. Auditing and Accounting

Auditors use the hypergeometric distribution to determine sample sizes and evaluate the probability of detecting errors in financial statements. It helps in designing sampling procedures that provide reliable estimates of error rates in larger populations of transactions.

3. Ecology and Population Studies

Ecologists use this distribution in mark-and-recapture studies to estimate wildlife population sizes. When studying a population where a known number of animals have been tagged, the hypergeometric distribution can model the probability of finding a certain number of tagged animals in subsequent samples.

4. Genetics and Biology

In genetics, the hypergeometric distribution is used to analyze gene sampling and to calculate the probability of drawing specific allele combinations. It's particularly useful in studying genetic drift and population genetics where sampling occurs without replacement.

5. Election and Survey Sampling

When conducting exit polls or analyzing voting patterns, the hypergeometric distribution helps calculate the probability of selecting voters with specific characteristics from a known population. This is especially useful in predicting election outcomes based on partial counts.