Hypergeometric Calculator Cards

The hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K successes.

What is Hypergeometric Distribution?

The hypergeometric distribution is used in scenarios where sampling is done without replacement from a finite population. It's commonly applied in quality control, genetic studies, and sampling problems.

Key characteristics of the hypergeometric distribution include:

Finite population size (N)
Number of successes in population (K)
Sample size (n)
Number of observed successes (k)

Unlike the binomial distribution, which assumes sampling with replacement, the hypergeometric distribution accounts for the decreasing probability of success as items are removed from the population.

How to Use the Calculator

To use the hypergeometric calculator:

Enter the total population size (N)
Enter the number of success states in the population (K)
Enter the sample size (n)
Enter the number of observed successes (k)
Click "Calculate" to get the probability

The calculator will display the probability of observing exactly k successes in a sample of size n from a population of size N with K successes.

Hypergeometric Formula

The probability mass function for the hypergeometric distribution is:

P(X = k) = [C(K, k) × C(N-K, n-k)] / C(N, n)

Where:

C(a, b) is the combination of a items taken b at a time
N = total population size
K = number of success states in population
n = sample size
k = number of observed successes

The calculator uses this formula to compute the exact probability of observing k successes in a sample of size n from a finite population.

Worked Examples

Example 1: Quality Control

A factory produces 1000 widgets, of which 50 are defective. A quality inspector randomly selects 10 widgets. What is the probability that exactly 2 are defective?

Using the calculator:

N = 1000
K = 50
n = 10
k = 2

The calculator would compute this probability using the hypergeometric formula.

Example 2: Genetic Studies

A genetic study involves 500 plants, of which 200 have a particular gene. Researchers select 20 plants. What is the probability that exactly 5 have the gene?

Using the calculator:

N = 500
K = 200
n = 20
k = 5

The calculator would compute this probability using the hypergeometric formula.

FAQ

When should I use the hypergeometric distribution instead of the binomial distribution?: Use the hypergeometric distribution when sampling is done without replacement from a finite population. The binomial distribution assumes sampling with replacement.
What happens if the sample size is larger than the population size?: The probability becomes zero because you cannot draw more items than exist in the population.
Can the hypergeometric distribution be used for continuous variables?: No, the hypergeometric distribution is specifically for discrete variables representing counts of successes in finite populations.
What are the assumptions of the hypergeometric distribution?: The assumptions are that the population is finite, the draws are without replacement, and the population contains exactly K successes.
How does the hypergeometric distribution relate to the binomial distribution?: The hypergeometric distribution reduces to the binomial distribution when the population size approaches infinity (N → ∞) and the sample size is small relative to the population size.