Bernoulli trial: Definition from Answers.com

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?
Were 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?
Did an employee 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.

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

$X(\omega) = \begin{cases} 1 & \mbox{if } \omega = s \\ 0 & \mbox{if } \omega = f. \end{cases}$

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

$E[X] = 1 \cdot p + 0 \cdot (1 - p) = p \,$

$V[X] = E[X^2] - E^2[X] = p - p^2 = p(1 - p) \,$

The standard deviation of X is simply $\sqrt{p(1-p)}.\,$

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".

	Bernoulli distribution
	Binomial Distribution
	Bernoulli sampling

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		Dictionary. The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2007. Published by Houghton Mifflin Company. All rights reserved. Read more
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Bernoulli trial

Mathematical description

See also