Hypergeometric Distribution

Hypergeometric Distribution (Table of Contents)

• Formula
• Examples

Hypergeometric Distribution Definition

The term “hypergeometric distribution” refers to the probability distribution of hypergeometric random variable which is primarily used to calculate probabilities when sampling without replacement. The formula for the probability of a hypergeometric distribution is derived using a number of items in the population, number of items in the sample, number of successes in the population, number of successes in the sample and few combinations. Mathematically, the probability is represented as,

where,

• N = No. of items in the population
• n = No. of items in the sample
• K = No. of successes in the population
• k = No. of successes in the sample

The mean and standard deviation of a hypergeometric distribution is expressed as,

Explanation of the Hypergeometric Distribution Formula

The equation for hypergeometric distribution can be derived by using the following steps:

Step 1: Firstly, determine the total number of items in the population which is denoted by N. For example, the number of playing cards in a deck which is 52.

Step 2: Next, determine the number of items in the sample which is denoted by n. For example, the number of cards drawn from the deck.

Step 3: Next, determine the instances which will be considered to be successes in the population and it is denoted by K. For example, the number of hearts in the overall deck which is 13.

Step 4: Next, determine the instances which will be considered to be successes in the sample drawn and it is denoted by k. For example, the number of hearts in the cards drawn from the deck.

Step 5: Finally, the formula for probability of a hypergeometric distribution is derived using number of items in the population (step 1), number of items in the sample (step 2), number of successes in the population (step 3) and number of successes in the sample (step 4) as shown below.

P = _K C _k * _{(N – K)} C _{(n – k)} / _N C _n

Examples of Hypergeometric Distribution Formula (with Excel Template)

Let’s see some simple to advanced examples of hypergeometric distribution equation to understand it better.

Example #1

Let us take the example of an ordinary deck of playing cards form where 6 cards are drawn randomly without replacement. Determine the probability of drawing exactly 4 red suites cards i.e., diamonds or hearts.

• Given, N = 52 (since there are 52 cards in an ordinary playing deck)
• n = 6 (Number of cards drawn randomly from the deck)
• K = 26 (since there are 13 red cards each in diamonds and hearts suite)
• k = 4 (Number of red cards to be considered successful in the sample drawn)

Solution:

Therefore, the probability of drawing exactly 4 red suites cards in the drawn 6 cards can be calculated using the above formula as,

Probability = _K C _k * _{(N – K)} C _{(n – k)} / _N C _n

= ₂₆ C ₄ * _{(52 – 26)} C _{(6 – 4)} / ₅₂ C ₆

= ₂₆ C ₄ * ₂₆ C ₂ / ₅₂ C ₆

= 14950 * 325 / 20358520

Probability will be –

Probability = 0.2387 ~ 23.87%

Therefore, there is a 23.87% probability of drawing exactly 4 red cards while drawing 6 random cards from an ordinary deck.

Example #2

Let us take another example of a wallet that contains 5 $100 bills and 7 $1 bills. If 4 bills are chosen randomly, then determine the probability of choosing exactly 3 $100 bills.

• Given, N = 12 (Number of $100 bills + Number of $1 bills)
• n = 4 (Number of bills chosen randomly)
• K = 5 (since there are 5 $100 bills)
• k = 3 (Number of $100 bills to be considered success in the sample chosen)

Solution:

Therefore, the probability of choosing exactly 3 $100 bills in the randomly chosen 4 bills can be calculated using the above formula as,

Probability = _K C _k * _{(N – K)} C _{(n – k)} / _N C _n

= ₅ C ₃ * _{(12 – 5)} C _{(4 – 3)} / ₁₂ C ₄

= ₅ C ₃ * ₇ C ₁ / ₁₂ C ₄

= 10 * 7 / 495

Probability will be –

Probability = 0.1414 ~ 14.14%

Therefore, there is 14.14% probability of choosing exactly 3 $100 bills while drawing 4 random bills.

Relevance and Uses

The concept of hypergeometric distribution is important because it provides an accurate way of determining the probabilities when the number of trials is not a very large number and that samples are taken from a finite population without replacement. In fact, the hypergeometric distribution is analogous to the binomial distribution which is used when the number of trials is substantially large. However, hypergeometric distribution is predominantly used for sampling without replacement.

Recommended Articles

This has been a guide to Hypergeometric Distribution Formula. Here we discuss how to calculate the probability of hypergeometric distribution in excel with examples and downloadable excel template. You can learn more about excel modeling from the following articles-

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