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
- <math>\sqrt{p(1-p)}.\,</math>
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".
