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.

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.

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.

Find [_{k}C_{x}]

N=10, n=6, k=5 and x=2
[_{k}C_{x}] = ( k! / (k-x)!) / x!
= (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! / 1!) / 4!) = 5 / 4! = 5.

Find [_{N}C_{n}]

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

This tutorial will guide you to calculate the hypergeometric distribution problems.