Hypergeometric Distribution Formula

Hypergeometric Distribution Formula (Table of Contents)

• Formula
• Examples

What is Hypergeometric Distribution Formula?

The hypergeometric distribution is basically a discrete probability distribution in statistics. It is very similar to binomial distribution and we can say that with confidence that binomial distribution is a great approximation for hypergeometric distribution only if the 5% or less of the population is sampled. If we have random draws, hypergeometric distribution is a probability of successes without replacing the item once drawn. But in a binomial distribution, the probability is calculated with replacement. For example, You have a basket which has N balls out of which “n” are black and you draw “m” balls without replacing any of the balls. So hypergeometric distribution is the probability distribution of the number of black balls drawn from the basket.

Formula For Hypergeometric Distribution:

Where,

• K – Number of “successes” in Population
• k – Number of “successes” in the sample
• N – Population size
• n – Sample Size

To understand the formula of hypergeometric distribution, one should be well aware of the binomial distribution and also with the Combination formula.

Combination Formula:

C(n,r) = n! / (r! * (n-r)!)

• n! – n factorial = n*(n-1)*(n-2)………..*1
• r! – r factorial = r*(r-1)*(r-2)………..*1
• (n-r)! – (n-r) factorial = (n-r)*(n-r-1)*(n-r-2)………..*1

Examples of Hypergeometric Distribution Formula (With Excel Template)

Let’s take an example to understand the calculation of the Hypergeometric Distribution in a better manner.

Hypergeometric Distribution Formula – Example #1

Let Say you have a deck of colored cards which has 30 cards out of which 12 are black and 18 are yellow. You have drawn 5 cards randomly without replacing any of the cards. Now you want to find the probability of exactly 3 yellow cards is drawn.

Solution:

Hypergeometric Distribution is calculated using the formula given below

Probability of Hypergeometric Distribution = C(K,k) * C((N – K), (n – k)) / C(N,n)

• Probability of getting exactly 3 yellow cards = C(18,3) * C((30-18), (5-3)) / C(30,5)
• Probability of getting exactly 3 yellow cards = C(18,3) * C(12, 2) / C(30,5)
• Probability of getting exactly 3 yellow cards = (18! / (3! * 15!) ) * (12! / (2! *10!) ) / (30! / (5! *25!) )
• Probability of getting exactly 3 yellow cards = 0.3779

Hypergeometric Distribution Formula – Example #2

Let say you live in a very small town which has 75 females and 95 males. Now there was voting which took place in your town and everyone voted. A sample of 20 voters was selected randomly. You want to calculate what is the probability that exactly 12 of these voters were male voters.

Solution:

Hypergeometric Distribution is calculated using the formula given below

Probability of Hypergeometric Distribution = C(K,k) * C((N – K), (n – k)) / C(N,n)

• Probability of getting 12 male voters = C(95,12) * C((170-95), (20-12)) / C(170,20)
• Probability of getting 12 male voters = C(95,12) * C(75, 8) / C(170,20)
• Probability of getting 12 male voters = (95! / (12! * 83!) ) * (75! / (8! *63!) ) / (170! / (20! *150!) )
• Probability of getting 12 male voters = 0.1766

Explanation

As discussed above, hypergeometric distribution is a probability of distribution which is very similar to a binomial distribution with the difference that there is no replacement allowed in the hypergeometric distribution. In order to perform this type of experiment or distribution, there are several criteria which need to be met.

• First and the foremost requirement is that data collected should be discrete in nature.
• Each pick or draw should not be replaced by another because whenever a random variable is drawn without replacement, then it is not independent and has relation to what is drawn earlier.
• There must be 2 sets of different group and you want to know the probability of a specific number of members of one group. For example, in the voting example, we have male and females. In bag example, we have a yellow and black group.

Along with these assumptions, knowledge of combination also plays a vital role in performing hypergeometric distribution. So it is imperative that one should know the concepts of combination before proceeding to hypergeometric distribution.

Relevance and Uses of Hypergeometric Distribution Formula

Hypergeometric distribution has many uses in statistics and in practical life. The most common use of the hypergeometric distribution, which we have seen above in the examples, is calculating the probability of samples when drawn from a set without replacement. In real life, the best example is the lottery. So in a lottery, once the number is out, it cannot go back and can be replaced, so hypergeometric distribution is perfect for this type of situations.

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This is a guide to Hypergeometric Distribution Formula. Here we discuss How to Calculate Hypergeometric Distribution along with practical examples. We also provide a downloadable excel template. You may also look at the following articles to learn more –

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