# Geometric Distribution - Definition, Formula, Mean, Examples

Probability theory is ac crucial department of mathematics that deals with the study of random events. One of the important concepts in probability theory is the geometric distribution. The geometric distribution is a discrete probability distribution that models the amount of trials needed to obtain the initial success in a series of Bernoulli trials. In this article, we will talk about the geometric distribution, extract its formula, discuss its mean, and provide examples.

## Definition of Geometric Distribution

The geometric distribution is a discrete probability distribution which describes the number of trials required to reach the first success in a sequence of Bernoulli trials. A Bernoulli trial is a trial which has two likely results, generally referred to as success and failure. For instance, tossing a coin is a Bernoulli trial since it can either come up heads (success) or tails (failure).

The geometric distribution is utilized when the tests are independent, meaning that the result of one test doesn’t impact the result of the next trial. Additionally, the probability of success remains constant across all the trials. We could signify the probability of success as p, where 0 < p < 1. The probability of failure is then 1-p.

## Formula for Geometric Distribution

The probability mass function (PMF) of the geometric distribution is specified by the formula:

P(X = k) = (1 - p)^(k-1) * p

Where X is the random variable which depicts the amount of test needed to achieve the first success, k is the number of trials required to achieve the first success, p is the probability of success in a single Bernoulli trial, and 1-p is the probability of failure.

Mean of Geometric Distribution:

The mean of the geometric distribution is described as the expected value of the amount of test required to achieve the first success. The mean is stated in the formula:

μ = 1/p

Where μ is the mean and p is the probability of success in a single Bernoulli trial.

The mean is the expected count of trials required to achieve the first success. For instance, if the probability of success is 0.5, then we expect to get the first success after two trials on average.

## Examples of Geometric Distribution

Here are some primary examples of geometric distribution

Example 1: Tossing a fair coin until the first head turn up.

Suppose we toss an honest coin until the first head turns up. The probability of success (getting a head) is 0.5, and the probability of failure (obtaining a tail) is as well as 0.5. Let X be the random variable that portrays the count of coin flips needed to get the first head. The PMF of X is stated as:

P(X = k) = (1 - 0.5)^(k-1) * 0.5 = 0.5^(k-1) * 0.5

For k = 1, the probability of getting the first head on the first flip is:

P(X = 1) = 0.5^(1-1) * 0.5 = 0.5

For k = 2, the probability of obtaining the initial head on the second flip is:

P(X = 2) = 0.5^(2-1) * 0.5 = 0.25

For k = 3, the probability of obtaining the initial head on the third flip is:

P(X = 3) = 0.5^(3-1) * 0.5 = 0.125

And so forth.

Example 2: Rolling an honest die until the first six shows up.

Suppose we roll an honest die till the initial six appears. The probability of success (obtaining a six) is 1/6, and the probability of failure (obtaining any other number) is 5/6. Let X be the irregular variable that depicts the count of die rolls needed to achieve the initial six. The PMF of X is given by:

P(X = k) = (1 - 1/6)^(k-1) * 1/6 = (5/6)^(k-1) * 1/6

For k = 1, the probability of obtaining the first six on the first roll is:

P(X = 1) = (5/6)^(1-1) * 1/6 = 1/6

For k = 2, the probability of obtaining the first six on the second roll is:

P(X = 2) = (5/6)^(2-1) * 1/6 = (5/6) * 1/6

For k = 3, the probability of obtaining the first six on the third roll is:

P(X = 3) = (5/6)^(3-1) * 1/6 = (5/6)^2 * 1/6

And so on.

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The geometric distribution is a crucial concept in probability theory. It is utilized to model a broad range of real-world phenomena, for instance the count of tests needed to obtain the first success in several scenarios.

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