Bernoulli distribution

The Bernoulli distribution models a single trial with two outcomes, success (probability p) or failure (probability 1 − p), with mean p and variance p(1 − p).

ByHoang TruongUpdated July 11, 2026

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4% of invoices processed contain an error, so p = 0.04. Modelled as a Bernoulli trial, an invoice splits into a 4% chance of error and a 96% chance of no error. The mean is E(X) = p = 0.04, and the variance is p(1 − p) = 0.04 × 0.96 = 0.0384.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

P(X=x) = p^x(1-p)^{1-x}, \quad x \in \{0,1\}

Variables

$p$: Probability of success on the trial
$x$: Outcome of the trial (1 = success, 0 = failure)

Gives the probability of either outcome of a single Bernoulli trial.

LaTeX

E[X]=p, \quad \operatorname{Var}(X)=p(1-p)

Variables

$p$: Probability of success on the trial
$E [X]$: Mean (expected value) of X
$Var (X)$: Variance of X

Gives the mean and variance of a Bernoulli random variable from its success probability alone.

Check yourself

PracticeCORE

A single website visitor either clicks an advertisement (success) or does not, with click probability p = 0.2. This is modelled as a Bernoulli random variable X. What is the variance of X?

Select an answer to check your understanding.