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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, called "success" and "failure."

In practice it refers to a single event which can have one of two possible outcomes. You can phrase these events into "yes or no" questions. For example:

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 my product?
Did a citizen vote for a specific candidate?
Is this employee going to vote pro-union?
Has this person been abducted by aliens before?

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 for example we designate a six as "success" and everything else as 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.
Call the birth of a baby of one sex "success" and of the other sex "failure." (Take your pick.)

Mathematically, such a trial is modeled by a random variable which can take only two values, 0 and 1, with 1 being thought off 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

√(p(1 − p)).

A Bernoulli process consists of repeatedly performing independent but identical Bernoulli trials, for instance flipping a coin 10 times.

Topics: Bernoulli Trial, Bernoulli's Inequality, Bernoulli Number...

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