Definition:Bernoulli Trial

From ProofWiki
Jump to: navigation, search

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