# Definition:Bernoulli Trial

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## 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 $0$

- the classic
**heads**and**tails**.

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 $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.

## Source of Name

This entry was named for Jacob Bernoulli.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Bernoulli trial** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Bernoulli trial** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Bernoulli trial** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Bernoulli trial**