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. The mathematical formalization of the Bernoulli trial is known as the Bernoulli process. This article offers an elementary introduction to the concept, whereas the article on the Bernoulli process offers a more advanced treatment.
Since a Bernoulli trial has only two possible outcomes, it can be framed as some "yes or no" question. For example:
- Did the coin land heads?
- Was the newborn child a girl?
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, not in any moral judgement. 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.
Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes "success" and the other outcome "failure". Let be the probability of success in a Bernoulli trial. Then the probability of failure is given by
Random variables describing Bernoulli trials are often encoded using the convention that 1 = "success", 0 = "failure".
Closely related to a Bernoulli trial is a binomial experiment, which consists of a fixed number of statistically independent Bernoulli trials, each with a probability of success , and counts the number of successes. A random variable corresponding to a binomial is denoted by , and is said to have a binomial distribution. The probability of exactly successes in the experiment is given by:
- Where is a Binomial coefficient
Bernoulli trials may also lead to negative binomial distributions (which count the number of successes in a series of repeated Bernoulli trials until a specified number of failures are seen), as well as various other distributions.
Example: tossing coins
Consider the simple experiment where a fair coin is tossed four times. Find the probability that exactly two of the tosses result in heads.
For this experiment, let a heads be defined as a success and a tails as a failure. Because the coin is assumed to be fair, the probability of success is . Thus the probability of failure, , is given by
Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by:
- Bernoulli scheme
- Bernoulli sampling
- Bernoulli distribution
- Binomial distribution
- Binomial coefficient
- Binomial proportion confidence interval
- Poisson sampling
- Sampling design
- Coin flipping
- Jacob Bernoulli
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