# Definition:Bernoulli Trial

## Definition

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

**Success**and**failure**;

**True**and**False**;

- $1$ and $0$;

- The classic
**heads**and**tails**.

Formally, a **Bernoulli trial** is modelled by a probability space $\left({\Omega, \Sigma, \Pr}\right)$ such that:

- $\Omega = \left\{{a, b}\right\}$

- $\Sigma = \mathcal P \left({\Omega}\right)$

- $\Pr \left({a}\right) = p, \Pr \left({b}\right) = 1 - p$

where $0 \le p \le 1$.^{[1]}

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

## Source of Name

This entry was named for Jacob Bernoulli.

## Notes

- ↑ Some sources insist that $0 < p < 1$, but it can be useful in certain circumstances to include the condition when the outcome is certainty.