# Bernoulli experiment

A Bernoulli experiment (or Bernoulli trial) is the simplest non-trivial random experiment imaginable: it's an experiment of which the outcome is random and can be either of two possibilities: success and failure.

The standard example is tossing a coin.

## Expectation and Variance

If the occurrence of a success is encoded as 1 and the occurrence of a failure with 0, then a probability of a success of p (and therefore, of a failure as q=1-p) leads to an expectation of

$\mathbf{E}[X] = p \cdot 1 + q \cdot 0 = p$

The variance is $Var(X) = \mathbf{E}[(X - \mathbf{E}[X])^2] = p(1-p)^2 + q(0-p)^2 = pq$.

The sum of n Bernoulli experiments with the same probability of success p follows a Binomial distribution B(n,p).