If you ever wondered what is the probability of getting 5 heads in a row while tossing a coin, this geometric distribution calculator might be of help. This article will help you understand the geometric distribution formula and definition. It will also present you with some examples of geometric distribution.

What is the geometric probability distribution

Geometric distribution describes number of failures prior to one success. For example, you may be throwing a die until you get a result of 6. Geometric distribution lets you determine the probability of getting a 6 at the first throw, the second and so on.

One of the properties of geometric distribution is memorylessness. It means that the probability distribution of the upcoming results does not depend on how many failures you already got. The exponential distribution has the same property.

Geometric distribution examples

Some examples of geometric distribution include:

  • Throwing a die multiple times to get a result of 6;
  • A couple that plans to have multiple children until the first boy;
  • Transmission of a sequence of bits until the first error;
  • Interviewing voters until you find someone who voted for the same candidate as you did.

Geometric distribution formula

The formula for geometric distribution is

P = (1-p)^x * p


  • x is the number of failures before the first success;
  • p is the probability of achieving a success in one trial;
  • P is the geometric probability of getting a success after x failures.

You can also use our geometric distribution calculator to find the following values:

  • Mean (the expected value) is equal to μ = (1-p)/p
  • Variance is equal to σ² = (1-p)/p²
  • Standard deviation is equal to σ = √[(1-p)/p²]

How to use the geometric distribution calculator: an example

Let's analyze the example with die throwing. You are throwing a die until getting the result of six. What are the chances that you will get a 6 on your second throw?

  1. Determine the probability of success for one trial. For a die, it is equal to 1/6.
  2. Calculate how many failures you will have before a success. For a successful second throw, only one throw will be a failure.
  3. Calculate the geometric probability with the help of the equation above:

P = (1-p)^x * p

P = (1-1/6)^1 * 1/6 = 0.1389 = 13.89%

  1. You can also calculate what is the expected number of throws needed before you get a success, the variance and standard deviation. Make sure to check the results with the geometric distribution calculator!

Take a look at our binomial distribution calculator, too!

Bogna Haponiuk

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