Hypergeometric Distribution

Hypergeometric Experiments
A hypergeometric experiment is a statistical experiment that has the following properties:

A sample of size n is randomly selected without replacement from a population of N items.
In the population, k items can be classified as successes, and N - k items can be classified as failures.
Consider the following statistical experiment. You have an urn of 10 marbles - 5 red and 5 green. You randomly select 2 marbles without replacement and count the number of red marbles you have selected. This would be a hypergeometric experiment.

Note that it would not be a binomial experiment. A binomial experiment requires that the probability of success be constant on every trial. With the above experiment, the probability of a success changes on every trial. In the beginning, the probability of selecting a red marble is 5/10. If you select a red marble on the first trial, the probability of selecting a red marble on the second trial is 4/9. And if you select a green marble on the first trial, the probability of selecting a red marble on the second trial is 5/9.

Note further that if you selected the marbles with replacement, the probability of success would not change. It would be 5/10 on every trial. Then, this would be a binomial experiment.

Notation
The following notation is helpful, when we talk about hypergeometric distributions and hypergeometric probability.

N: The number of items in the population.
k: The number of items in the population that are classified as successes.
n: The number of items in the sample.
x: The number of items in the sample that are classified as successes.
kCx: The number of combinations of k things, taken x at a time.
h(x; N, n, k): hypergeometric probability - the probability that an n-trial hypergeometric experiment results in exactly x successes, when the population consists of N items, k of which are classified as successes.
Hypergeometric Distribution
A hypergeometric random variable is the number of successes that result from a hypergeometric experiment. The probability distribution of a hypergeometric random variable is called a hypergeometric distribution.

Given x, N, n, and k, we can compute the hypergeometric probability based on the following formula:

Hypergeometric Formula. Suppose a population consists of N items, k of which are successes. And a random sample drawn from that population consists of n items, x of which are successes. Then the hypergeometric probability is:
h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]

The hypergeometric distribution has the following properties:

The mean of the distribution is equal to n * k / N .
The variance is n * k * ( N - k ) * ( N - n ) / [ N2 * ( N - 1 ) ] .
Example 1

Suppose we randomly select 5 cards without replacement from an ordinary deck of playing cards. What is the probability of getting exactly 2 red cards (i.e., hearts or diamonds)?

Solution: This is a hypergeometric experiment in which we know the following:

N = 52; since there are 52 cards in a deck.
k = 26; since there are 26 red cards in a deck.
n = 5; since we randomly select 5 cards from the deck.
x = 2; since 2 of the cards we select are red.
We plug these values into the hypergeometric formula as follows:

h(x; N, n, k) = [ kCx ] [ N-kCn-x ] / [ NCn ]
h(2; 52, 5, 26) = [ 26C2 ] [ 26C3 ] / [ 52C5 ]
h(2; 52, 5, 26) = [ 325 ] [ 2600 ] / [ 2,598,960 ] = 0.32513
Thus, the probability of randomly selecting 2 red cards is 0.32513.