# Bernoulli trial

## Any experiment with two possible random outcomes / From Wikipedia, the free encyclopedia

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In the theory of probability and statistics, a **Bernoulli trial** (or **binomial trial**) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.^{[1]} It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his * Ars Conjectandi* (1713).

^{[2]}

The mathematical formalization and advanced formulation of the Bernoulli trial is known as the Bernoulli process.

Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example:

- Is the top card of a shuffled deck an ace?
- Was the newborn child a girl? (See human sex ratio.)

Therefore, success and failure are merely labels for the two outcomes, and should not be construed literally. The term "success" in this sense consists in the result meeting specified conditions; it is not a value judgement. More generally, given any probability space, for any event (set of outcomes), one can define a Bernoulli trial, corresponding to whether the event occurred or not (event or complementary event). 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. In this case, there are exactly two possible outcomes.
- Rolling a die, where a six is "success" and everything else a "failure". In this case, there are six possible outcomes, and the event is a six; the complementary event "not a six" corresponds to the other five possible outcomes.
- In conducting a political opinion poll, choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.