Hypergeometric Distribution Definition

In the statistics and the probability theory, hypergeometric distribution is basically a distinct probability distribution which defines probability of k successes (i.e. some random draws for the object drawn that has some specified feature) in n no of draws, without any replacement, from a given population size N which includes accurately K objects having that feature, where the draw may succeed or may fail.

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,

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

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For eg:
Source: Hypergeometric Distribution (wallstreetmojo.com)

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,

Mean = n * K / N

Standard Deviation = [n * K * (N – K) * (N – n) / {N² * (N – 1)}]^1/2

Explanation

Follow the below steps:

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 is 52.
Next, determine the number of items in the sample, denoted by n—for example, the number of cards drawn from the deck.
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.
Next, determine the instances which will be considered to be successes in the sample drawn, and it is denoted by k. E.g., the number of hearts in the cards drawn from the deck.
Finally, the formula for the probability of a hypergeometric distribution is derived using a 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 = _KC _k* _{(N – K)}C _{(n – k)}/_NC _n

Examples of Hypergeometric Distribution (with Excel Template)

You can download this Hypergeometric Distribution Excel Template here – Hypergeometric Distribution Excel Template

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,

Hypergeometric Distribution Formula Example1

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

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

= ₂₆C ₄* ₂₆C ₂/₅₂C ₆

= 14950 * 325 / 20358520

The 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 a 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,

Hypergeometric Distribution Formula Example2

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

= ₅C ₃* _{(12 – 5)}C _{(4 – 3)}/₁₂C ₄

= ₅C ₃* ₇C ₁/₁₂C ₄

= 10 * 7 / 495

Probability will be –

Probability = 0.1414 ~ 14.14%

Therefore, there is a 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 distributionBinomial Distribution The Binomial Distribution Formula calculates the probability of achieving a specific number of successes in a given number of trials. nCx represents the number of successes, while (1-p) n-x represents the number of trials.read more, which is used when the number of trials is substantially large. However, hypergeometric distribution is predominantly used for sampling without replacement.

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Hypergeometric Distribution