A **geometric distribution** tells us how many trials (*k*) are required until we obtain the first “success.”

The probability that we will obtain the first success on the k^{th} trial can be found using the formula:

P(obtain first success on k^{th} trial) = (1-p)^{k-1} * p

where *p *is the probability of success in a given trial.

The geometric distribution has the following properties:

- The mean of the distribution is
**μ**= 1/p - The variance of the distribution is
**σ**= (1-p) / p^{2}^{2} - The standard deviation of the distribution is
**σ**= √σ^{2}

Let’s walk through some examples to get a better understanding of the geometric distribution.

**Examples Using the Geometric Distribution**

**Example 1: An ice cream shop gives a raffle ticket to each customer that walks in the door. If a customer receives a winning raffle ticket, they get a free ice cream cone. The probability that a given raffle ticket is a winner is 5%. What is the probability that the 10^{th} customer that walks in the door is the first winner?**

**Step 1: Identify the probability of success on a given trial (p) and the trial we are interested in studying (k).**

The probability that a given raffle ticket is a winner is *p *= 0.05

We want to know if the *k *= 10^{th} customer will be the first winner.

**Step 2: Plug these numbers into the geometric formula or a geometric calculator.**

**Using the formula:**

P(10^{th} customer is the first winner) = (1-p)^{k-1} * p

P(10^{th} customer is the first winner) = (1-.05)^{10-1} * .05 = **.03151**

**Using the calculator:**

Plug the following numbers into the Geometric Distribution Calculator:

The probability that the 10^{th} customer who walks into the ice cream shop will be the first winner is **.03151**.

**Example 2: Amanda makes 70% of her free-throw attempts. What is the probability that she makes her first free-throw within her first two attempts?**

**Step 1: Identify the probability of success on a given trial (p) and the trial we are interested in studying (k).**

The probability that Amanda makes a given free-throw is p = 0.7

We want to know if she will make her *k *= 1^{st} or *k *= 2^{nd }attempt.

**Step 2: Plug these numbers into the geometric formula or a geometric calculator.**

**Using the formula:**

P(makes the 1^{st} attempt) = (1-0.7)^{1-1} * 0.7 = 0.7

P(makes the 2^{nd }attempt) = (1-0.7)^{2-1} * 0.7 = 0.21

P(makes the 1^{st} or 2^{nd }attempt) = 0.7 + 0.21 = **0.91**

**Using the calculator:**

Plug the following numbers into the Geometric Distribution Calculator:

The probability that Amanda makes her first free-throw within her first two attempts is** 0.91**.

**Example 3: Max makes 80% of his free-throw attempts. What is the probability that he needs to shoot three or more free-throws to make his first shot?**

**Using the calculator:**

Plug the following numbers into the Geometric Distribution Calculator:

The probability that Max needs to shoot three or more free-throws to make his first shot is **0.04**.