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.

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.

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.

h(x;N;n;k) = [_{k}C_{x}] [_{N - k}C_{n - x}] / [_{N}C_{n}]

Steps to find hypergeometric probability function

Find [_{k}C_{x}]

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

[_{k}C_{x}] = ( k! / (k - x)!) / x!

_{5}C_{2} is the probability of selecting two red balls from 5 random balls

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

Find [_{N-k}C_{n-x}]

N - k=5 and n - x=4
[_{N - k}C_{n - x}] = ((N - k)! / ((N - k) - (n - x))!) / (n - x)!

_{5}C_{4} is the probability of selecting 4 red balls from a random 5 balls.

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

Find [_{N}C_{n}]

N=10 and n=6
[_{N}C_{n}] = ( N! / (N - n)!) / n!)

_{10}C_{6} is the probability of selecting 6 red balls from the total 10 balls

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

Find [_{k}C_{x}] [_{N - k}C_{n - x}] / [_{N}C_{n}]

[_{k}C_{x}] = 10, [_{N - k}C_{n - x}] = 5 and [_{N}C_{n}] = 210.

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

h(x;N;n;k) = [_{k}C_{x}] [_{N - k}C_{n - x}] / [_{N}C_{n}]
= [_{5}C_{2}] [_{5}C_{4}] / [_{10}C_{6}]
= (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.