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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
${h(x;N,n,K) = frac{[C(k,x)][C(N-k,n-x)]}{C(N,n)}}$
Where −
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${N}$ = items in the population
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${k}$ = successes in the population.
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${n}$ = items in the random sample drawn from that population.
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${x}$ = 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:
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N = 52; since there are 52 cards in a deck.
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k = 26; since there are 26 red cards in a deck.
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n = 5; since we randomly select 5 cards from the deck.
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x = 2; since 2 of the cards we select are red.
We plug these values into the hypergeometric formula as follows:
h(2; 52, 5, 26) = frac{[C(26,2)][C(52-26,5-2)]}{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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