Statistics – Hypergeometric Distribution
A hypergeometric random variable is the number of successes that
result from a hypergeometric experiment. The probability
distribution of a hypergeometric random variable is called a
hypergeometric distribution.
Hypergeometric distribution is defined and given by the following
probability function:
Formula
${h(x;N,n,K) = frac{[C(k,x)][C(Nk,nx)]}{C(N,n)}}$
Where −

${N}$ = items in the population

${k}$ = successes in the population.

${n}$ = items in the random sample drawn from that
population. 
${x}$ = successes in the random sample.
Example
Problem Statement:
Suppose we randomly select 5 cards without replacement from an
ordinary deck of playing cards. What is the probability of
getting exactly 2 red cards (i.e., hearts or diamonds)?
Solution:
This is a hypergeometric experiment in which we know the
following:

N = 52; since there are 52 cards in a deck.

k = 26; since there are 26 red cards in a deck.

n = 5; since we randomly select 5 cards from the deck.

x = 2; since 2 of the cards we select are red.
We plug these values into the hypergeometric formula as follows:
52, 5, 26) = frac{[C(26,2)][C(5226,52)]}{C(52,5)} \[7pt] =
frac{[325][2600]}{2598960} \[7pt] = 0.32513 }$
Thus, the probability of randomly selecting 2 red cards is
0.32513.
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