Probability Distribution Function : The Hypergeometric Distribution

4 min readFeb 26, 2020

Hypergeometric Distribution

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

Where −

· N = items in the population

· k = successes in the population.

· n = items in the random sample drawn from that population.

· 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:

· N = 52; since there are 52 cards in a deck.

· k = 26; since there are 26 red cards in a deck.

· n = 5; since we randomly select 5 cards from the deck.

· x = 2; since 2 of the cards we select are red.

We plug these values into the hypergeometric formula as follows:

Thus, the probability of randomly selecting 2 red cards is 0.32513.

here are five characteristics of a hypergeometric experiment.

1. You take samples from two groups.

2. You are concerned with a group of interest, called the first group.

3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of ten players.

4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a woman first is . The probability of picking a man second is if a woman was picked first. It is if a man was picked first. The probability of the second pick depends on what happened in the first pick.

5. You are not dealing with Bernoulli Trials.

Cumulative Hypergeometric Probability

A cumulative hypergeometric probability refers to the probability that the hypergeometric random variable is greater than or equal to some specified lower limit and less than or equal to some specified upper limit.

For example, suppose we randomly select five cards from an ordinary deck of playing cards. We might be interested in the cumulative hypergeometric probability of obtaining 2 or fewer hearts. This would be the probability of obtaining 0 hearts plus the probability of obtaining 1 heart plus the probability of obtaining 2 hearts, as shown in the example below.

Example 2

Suppose we select 5 cards from an ordinary deck of playing cards. What is the probability of obtaining 2 or fewer hearts?

Solution: This is a hyper geometric experiment in which we know the following:

N = 52$; since there are 52 cards in a deck.

k = 13$; since there are 13 hearts in a deck.

n = 5$; since we randomly select 5 cards from the deck.

x = 0$ to 2; since our selection includes 0, 1, or 2 hearts.

These values into the hyper geometric formula as follows:

h(x < x; N, n, k) = h(x < 2; 52, 5, 13)$

h(x < 2; 52, 5, 13) = h(x = 0; 52, 5, 13) + h(x = 1; 52, 5, 13) + h(x = 2; 52, 5, 13)$

h(x < 2; 52, 5, 13) = [ (13C0) (39C5) / (52C5) ] + [ (13C1) (39C4) / (52C5) ]
+ [ (13C2) (39C3) / (52C5) ]

h(x < 2; 52, 5, 13) = [ (1)(575,757)/(2,598,960) ] + [ (13)(82,251)/(270,725) ]
+ [ (78)(9139)/(22,100) ]

h(x < 2; 52, 5, 13) = [ 0.2215 ] + [ 0.4114 ] + [ 0.2743 ] $

h(x < 2; 52, 5, 13) = 0.9072

Thus, the probability of randomly selecting at most 2 hearts is 0.9072.

Binomial Distribution

Binomial appropriation is a discrete likelihood conveyance. This distribution was discovered by a Swiss Mathematician James Bernoulli. It is used in such situation where an experiment results in two possibilities — success and failure. Binomial distribution is a discrete probability distribution which expresses the probability of one set of two alternatives-successes (p) and failure (q). Binomial distribution is defined and given by the following probability function: