Bernoulli trial
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. It is named after Jacob Bernoulli, a 17th-century Swiss mathematician, who analyzed them in his "Ars Conjectandi" (1713). The mathematical formalisation 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: 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. 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: Definition. 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 formula_1 be the probability of success in a Bernoulli trial, and formula_2 be the probability of failure. Then the probability of success and the probability of failure sum to one, since these are complementary events: "success" and "failure" are mutually exclusive and exhaustive. Thus one has the following relations: Alternatively, these can be stated in terms of odds: given probability "p" of success and "q" of failure, the "odds for" are formula_4 and the "odds against" are formula_5 These can also be expressed as numbers, by  dividing, yielding the odds for, formula_6, and the odds against, formula_7, These are multiplicative inverses, so they multiply to 1, with the following relations: In the case that a Bernoulli trial is representing an event from finitely many equally likely outcomes, where "S" of the outcomes are success and "F" of the outcomes are failure, the odds for are formula_10 and the odds against are formula_11 This yields the following formulas for probability and odds: Note that here the odds are computed by dividing the number of outcomes, not the probabilities, but the proportion is the same, since these ratios  only differ by multiplying both terms by the same constant factor. 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 formula_13 of statistically independent Bernoulli trials, each with a probability of success formula_1, and counts the number of successes. A random variable corresponding to a binomial experiment is denoted by formula_15, and is said to have a "binomial distribution". The probability of exactly formula_16 successes in the experiment formula_15 is given by: where formula_19 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. When multiple Bernoulli trials are performed, each with its own probability of success, these are sometimes referred to as Poisson trials. 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 formula_20. Thus the probability of failure, formula_2, 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: