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The geometric distribution describes the probability of experiencing a certain amount of failures before experiencing the first success in a series of Bernoulli trials.

A

Bernoulli trialis an experiment with only two possible outcomes – “success” or “failure” – and the probability of success is the same each time the experiment is conducted.

An example of a Bernoulli trial is a coin flip. The coin can only land on two sides (we could call heads a “success” and tails a “failure”) and the probability of success on each flip is 0.5, assuming the coin is fair.

If a random variable *X* follows a geometric distribution, then the probability of experiencing *k *failures before experiencing the first success can be found by the following formula:

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

where:

**k:**number of failures before first success**p:**probability of success on each trial

The following examples show how to calculate probabilities related to the geometric distribution in Excel.

**Example 1: Flipping a Coin**

Suppose we are flipping a coin and we want to know the probability that it will take exactly three “failures” until a coin finally lands on heads.

We would use the following formula to calculate this probability:

The probability that we will experience three “failures” until a coin finally lands on heads is **.0625**.

**Example 2: Shooting Free Throws**

Suppose a certain basketball player makes 60% of his free throws. What is the probability that the player will miss four free throws until he finally makes one?

We would use the following formula to calculate this probability:

The probability that the player will miss four free throws until he finally makes one is **.01536**.

**Example 3: Supporting a Law**

Suppose a researcher is waiting outside of a library to ask people if they support a certain law. The probability that a given person supports the law is p = 0.2. What is the probability that the fourth person the researcher talks to is the first person to support the law?

We would use the following formula to calculate this probability:

The probability that the fourth person the researcher talks to is the first person to support the law is **.1024**.

**Additional Resources**

An Introduction to the Geometric Distribution

Geometric Distribution Calculator

5 Real-Life Examples of the Geometric Distribution