In the theory of probability

Probability

Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, 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

Experiment

An experiment is a methodical procedure carried out with the goal of verifying, falsifying, or establishing the validity of a hypothesis. Experiments vary greatly in their goal and scale, but always rely on repeatable procedure and logical analysis of the results...

 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?

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
  Fair coin
  In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin...
  
   has the probability of success 0.5 by definition.
• Rolling a die
  Dice
  A die is a small throwable object with multiple resting positions, used for generating random numbers...
  
  , where a six is "success" and everything else a "failure".
• In conducting a political opinion poll
  Opinion poll
  An opinion poll, sometimes simply referred to as a poll is a survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions of a population by conducting a series of questions and then extrapolating generalities in ratio or within confidence...
  
  , choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.

Definition

Independent repeated trials of an experiment with two outcomes only 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.

A binomial experiment consisting of a fixed number of trials, each with a probability of success , is denoted by .
The probability of exactly successes in the experiment is given by:.

The function for for is called a binomial distribution.

Bernoulli trials may also lead to negative binomial

Negative binomial distribution

In probability theory and statistics, the negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of Bernoulli trials before a specified number of failures occur...

, geometric, and other distributions as well.

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.

Solution

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:.

See also

• Bernoulli scheme
  Bernoulli scheme
  In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes are important in the study of dynamical systems, as most such systems exhibit a repellor that is the product of the Cantor set and a smooth...
  
  
• Bernoulli sampling
  Bernoulli sampling
  In the theory of finite population sampling, Bernoulli sampling is a sampling process where each element of the population that is sampled is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample during the drawing of a single sample...
  
  
• Bernoulli distribution
• Binomial distribution
• Binomial proportion confidence interval
  Binomial proportion confidence interval
  In statistics, a binomial proportion confidence interval is a confidence interval for a proportion in a statistical population. It uses the proportion estimated in a statistical sample and allows for sampling error. There are several formulas for a binomial confidence interval, but all of them rely...
  
  
• Poisson sampling
  Poisson sampling
  In the theory of finite population sampling, Poisson sampling is a sampling process where each element of the population that is sampled is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample during the drawing of a single sample.Each element...
  
  
• Sampling design
  Sampling design
  In the theory of finite population sampling, a sampling design specifies for every possible sample its probability of being drawn.Mathematically, a sampling design is denoted by the function P which gives the probability of drawing a sample S....
  
  
• Coin flipping
  Coin flipping
  Coin flipping or coin tossing or heads or tails is the practice of throwing a coin in the air to choose between two alternatives, sometimes to resolve a dispute between two parties...
  
  
• Jacob Bernoulli