# 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 $0$

- 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.

## 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

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Bernoulli trial**