The Bernoulli Distribution

A Bernoulli trial is a random experiment with exactly two possible outcomes, conventionally labeled success and failure. We let $p$ denote the probability of success, so the probability of failure is $1-p.$

A random variable $X$ follows a Bernoulli distribution with parameter $p,$ written $X \sim \text{Bernoulli}(p),$ if

$$P(X=1) = p, \qquad P(X=0) = 1-p.$$

Here $X=1$ records a success and $X=0$ records a failure. The two cases can be combined into a single probability mass function (PMF):

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

For example, flipping a fair coin and setting $X=1$ for heads and $X=0$ for tails gives $X \sim \text{Bernoulli}(0.5),$ with

$$P(X=1) = 0.5, \qquad P(X=0) = 0.5.$$