# 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

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