# Definition:Bernoulli Trial

## Definition

A Bernoulli trial is an experiment whose sample space has two elements, which can be variously described, for example, as:

Success and failure
True and False
$1$ and $0$

Formally, a Bernoulli trial is modelled by a probability space $\struct {\Omega, \Sigma, \Pr}$ such that:

$\Omega = \set {a, b}$
$\Sigma = \powerset \Omega$
$\map \Pr a = p, \map \Pr b = 1 - p$

where:

$\powerset \Omega$ denotes the power set of $\Omega$
$0 \le p \le 1$

That is, $\Pr$ obeys a Bernoulli distribution.

### Bernoulli Variable

Let $X$ be a discrete random variable whose sample space is $\Omega$ in such a Bernoulli trial.

Then $X$ is known as a Bernoulli variable.

## Also defined as

Some sources insist that the valid codomain of a Bernoulli trial is $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.

## Also see

• Results about Bernoulli trials can be found here.

## Source of Name

This entry was named for Jacob Bernoulli.

## Historical Note

The concept of a Bernoulli trial was first raised by Jacob Bernoulli in $1713$.