# 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, called "success" and "failure."

In practice it refers to a single event which can have one of two possible outcomes. These events can be phrased 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.
• Calling the birth of a baby of one sex "success" and of the other sex "failure."

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

[itex]\sqrt{p(1-p)}.\,[itex]

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

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