The geometric distribution is a discrete memoryless probability distribution which describes the number of failures before the first success, x. The term also commonly refers to a secondary probability distribution, which describes the number of trials with two possible outcomes, success or failure, up to and including until the first success, x. This is known as the shifted geometric distribution. In this application, we will look closer at the first distribution.

Given a sequence of Bernoulli trials where p is the probability of success on each trial, the probability function for the geometric distribution is:

$P\left(Y\=x\right) \= p {\left(1-p \right)}^x$, for x = 0, 1, 2, 3...

where x is the number of trials or failures before the first success.

The cumulative probability function is given by:

$1-{\left(1-p\right)}^{x\+1}$