# Bernoulli trial

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In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure".

In practice it refers to a single experiment which can have one of two possible outcomes. These events can be phrased into "yes or no" questions:

• Did the coin land heads?
• Was the newborn child a girl?

Therefore success and failure are labels for outcomes, and should not be construed literally. Examples of Bernoulli trials include

• Flipping a coin. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A fair coin has the probability of success 0.5 by definition.
• Rolling a die, where a six is "success" and everything else a "failure".
• In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.

### Mathematical description

Mathematically, a Bernoulli trial can be described by a sample space Ω consisting of two values, s for "success" and f for "failure". Therefore the sample space is $\Omega = \{s, f\} \,$. Then a random variable X can be defined on this sample space, that is, a function $X : \Omega \mapsto \mathbf{R}$. In this case the random variable is very simple and given by

If p is the probability of observing a 1 and 1 − p the probability of observing a 0 (the probability distribution of X), then X has a Bernoulli distribution and the expected value and the variance of X are given by

The standard deviation of X is $\sqrt{p(1-p)}.\,$ This distribution is a special case of the Binomial distribution.

A Bernoulli process consists of repeatedly performing independent but identical Bernoulli trials.

The process of determining an expectation value and deviation, based on a limited number of Bernoulli trials is colloquially known as "checking if a coin is fair".