Hypergeometric Distribution Definition

The probability of a hypergeometric distribution is derived using the 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  represents as,

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where,

  • N = No. of items in the populationn = No. of items in the sampleK = No. of successes in the populationk = No. of successes in the sample

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

Explanation

Examples of Hypergeometric Distribution (with Excel Template)

Example #1

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

  • 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 several items in the population (Step 1), the number of items in the sample (Step 2), the number of successes in the population (Step 3), and the number of successes in the sample (Step 4) as shown below.

  • 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 suit)k = 4 (Number of red cards to be considered successful in the sample drawn)

Solution:

Therefore, the probability of drawing exactly 4 red suits cards in the draw 6 cards calculated using the above formula,

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

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)

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

= 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 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 very large and when samples are taken from a finite population without replacement. The hypergeometric distribution is analogous to the binomial distributionBinomial DistributionThe 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, used when the number of trials is substantially large. However, hypergeometric distribution is predominantly used for sampling without replacement.

This article is a guide to Hypergeometric Distribution Formula. Here, we discuss calculating hypergeometric distribution probability in Excel with examples and a downloadable Excel template. You can learn more about Excel modeling from the following articles: –

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