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The hypergeometric distributionÂ describes the probability of choosingÂ *k *objects with a certain feature inÂ *n *draws without replacement, from a finite population of sizeÂ *NÂ *that containsÂ *KÂ *objects with that feature.

If aÂ random variableÂ *X*Â follows a hypergeometric distribution, then the probability of choosingÂ *kÂ *objects with a certain feature can be found by the following formula:

**P(X=k) = _{K}C_{k}Â (_{N-K}C_{n-k}) /Â _{N}C_{n}**

where:

**N:Â**population size**K:Â**number of objects in population with a certain feature**n:Â**sample size**k:Â**number of objects in sample with a certain featurenumber of combinations of K things taken k at a time_{K}C_{k}:Â

To calculate probabilities related to the hypergeometric distribution in Excel, we can use the following formula:

=HYPGEOM.DIST(sample_s, number_sample, population_s, number_pop, cumulative)

where:

**sample_s**: number of successes in sample**number_sample**: size of sample**population_s**: number of successes in population**number_pop**: population size**cumulative**: Whether to calculate the cumulative distribution function

The following examples show how to use this formula in practice.

**Example 1: Picking Cards from a Deck**

There are 4 Queens in a standard deck of 52 cards. Suppose we randomly pick a card from a deck, then, without replacement, randomly pick another card from the deck. What is the probability that both cards are Queens?

We can use the following formula in Excel to calculate the probability that both cards are Queens:

The probability that both cards are Queens is .**00452**.

**Example 2: Picking Balls from an Urn**

An urn contains 3 red balls and 5 green balls. You randomly choose 4 balls. What is the probability that you choose exactly 2 red balls?

We can use the following formula in Excel to find this probability:

The probability that you choose exactly 2 red balls is .**428571**.

**Example 3: Choosing Marbles from a Basket**

A basketÂ contains 7 purple marbles and 3 pink marbles. You randomly choose 6 marbles. What is the probability that you choose exactly 3 pink marbles?

We can use the following formula in Excel to find this probability:

The probability that you choose exactly 3 pink marbles is .**16667**.

**Additional Resources**

An Introduction to the Hypergeometric Distribution

Online Hypergeometric Distribution Calculator