Bernoulli trial
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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, "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.
[edit] 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 
. Then a random variable X can be defined on this sample space, that is, a function 
. In this case the random variable is very simple and given by
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 \,](http://upload.wikimedia.org/math/4/d/6/4d6bfd64cb8eddce974d3f077ef27200.png)
![V[X] = E[X^2] - E^2[X] = p - p^2 = p(1 - p) \,](http://upload.wikimedia.org/math/3/7/9/379e975d0b0b2e52a67a03199549c5e8.png)
The standard deviation of X is simply 
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".
