The Geometric Distribution

An explanation of the geometric distribution
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 kth trial can be found using the formula:

P(obtain first success on kth trial) = (1-p)k-1 * p

where 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 σ2 = (1-p) / p2
  • The standard deviation of the distribution is σ = √σ2

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 10th 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 = 0.05

We want to know if the = 10th customer will be the first winner.

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

Using the formula:

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

P(10th 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 10th 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 = 1st or = 2nd attempt.

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

Using the formula:

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

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

P(makes the 1st or 2nd 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.

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