# Hypergeometric Distribution Tutorial

The probability of success in a sample of elements drawn without repetition is called as hypergeometric probability distribution. It is also used in many mathematical functions. A simple online free hypergeometric distribution tutorial with formula and solved example.

## Hypergeometric Probability Distribution

##### Definition:

In statistics, hypergeometric distribution is one of the discrete probability distribution. This distribution is used for calculating the probability for a random selection of an object without repetition. Here, population size is the total number of objects in the experiment.

#### Formula:

h(x;N;n;k) = [kCx] [N-kCn-x] / [NCn] Where, N is the total population size n is the total sample size k is the number of selected items from the population size x is a random variable
##### Hypergeometric distribution Example:

Consider, 5 balls are chosen randomly from the total of 10 balls without repetition. Calculate the probability of getting exactly 2 red balls out of 6 red balls.

###### Formula

h(x;N;n;k) = [kCx] [N - kCn - x] / [NCn]

Steps to find hypergeometric probability function

Find [kCx]

###### where,

Total number of balls (N) = 10 Total number of red balls (n) = 6 Balls chosen randomly (k) = 5 Exactly two red balls chosen(x) = 2

[kCx] = ( k! / (k - x)!) / x!

5C2 is the probability of selecting two red balls from 5 random balls

= (5! / (5 - 2)!) / 2! = 20 / 2 = 10.

Find [N-kCn-x]

###### where,

N - k=5 and n - x=4 [N - kCn - x] = ((N - k)! / ((N - k) - (n - x))!) / (n - x)!

5C4 is the probability of selecting 4 red balls from a random 5 balls.

= ((5! / 1!) / 4!) = 5! / 4! = 5.

Find [NCn]

###### where,

N=10 and n=6 [NCn] = ( N! / (N - n)!) / n!)

10C6 is the probability of selecting 6 red balls from the total 10 balls

= ((10! / 4!) / 6!) = 151200 / 6! = 210.

###### Step4:

Find [kCx] [N - kCn - x] / [NCn]

###### where,

[kCx] = 10, [N - kCn - x] = 5 and [NCn] = 210.

Substitute the values in the formula to find the hypergeometric distribution function.

h(x;N;n;k) = [kCx] [N - kCn - x] / [NCn] = [5C2] [5C4] / [10C6] = (10 x 5) / 210 = 0.238.

Hence there are 23.8% possibilities for choosing exactly 2 red balls without repetition.

Solve different hypergeometric probability distribution problems using the above example and formula. This hypergeometric distribution tutorial will guide you to calculate the hypergeometric distribution function.