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.
N=10, n=6, k=5 and x=2 [kCx] = ( k! / (k-x)!) / x! = (5! / (5-2)!) / 2! = 20 / 2 = 10.
N-k=5 and n-x=4 [N-kCn-x] = ((N-k)! / ((N-k)-(n-x))!) / (n-x)! = ((5! / 1!) / 4!) = 5 / 4! = 5.
N=10 and n=6 [NCn] = ( N! / (N-n)!) / n!) = ((10! / 4!) / 6!) = 151200 / 6! = 210.
Find [kCx] [N-kCn-x] / [NCn]
[kCx] = 10, [N-kCn-x] = 5 and [NCn] = 210. 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.
This tutorial will guide you to calculate the hypergeometric distribution problems.