Bernoulli trial

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:


 * Will the coin land heads?
 * Was the newborn child a girl?
 * Are a person's eyes green?
 * Did a mosquito die after the area was sprayed with insecticide?
 * Did a potential customer decide to buy a product?
 * Did a citizen vote for a specific candidate?
 * Is this employee going to vote pro-union?

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.

Mathematically, such a trial is modeled by a random variable which can take only two values, 0 and 1, with 1 being thought of as "success". If p is the probability of success, then the expected value of such a random variable is p and its standard deviation is


 * $$\sqrt{p(1-p)}.\,$$

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".