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.
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.
Step1: Find [kCx]
where, N=10, n=6, k=5 and x=2
[kCx] = ( k! / (k-x)!) / x!
= (5! / (5-2)!) / 2! = 20 / 2 = 10.
Step2: Find [N-kCn-x]
where, N-k=5 and n-x=4
[N-kCn-x] = ((N-k)! / ((N-k)-(n-x))!) / (n-x)!
= ((5! / 1!) / 4!) = 5 / 4! = 5.
Step3: Find [NCn]
where, N=10 and n=6
[NCn] = ( N! / (N-n)!) / n!)
= ((10! / 4!) / 6!) = 151200 / 6! = 210.
Step4: Find [kCx] [N-kCn-x] / [NCn]
where,
[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.
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