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In the theory of
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
and
statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, a Bernoulli trial (or binomial trial) is a random
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into Causality, cause-and-effect by demonstrating what outcome oc ...
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 Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
, a 17th-century Swiss mathematician, who analyzed them in his ''
Ars Conjectandi (Latin for "The Art of Conjecturing") is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apa ...
'' (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: *Is the top card of a shuffled deck an ace? *Was the newborn child a girl? (See
human sex ratio In anthropology and demography, the human sex ratio is the ratio of males to females in a population. Like most sexual species, the sex ratio in humans is close to 1:1. In humans, the natural ratio at birth between males and females is sligh ...
.) 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 In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
, for any
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
(set of outcomes), one can define a Bernoulli trial, corresponding to whether the event occurred or not (event or
complementary event In probability theory, the complement of any event ''A'' is the event ot ''A'' i.e. the event that ''A'' does not occur.Robert R. Johnson, Patricia J. Kuby: ''Elementary Statistics''. Cengage Learning 2007, , p. 229 () The event ''A'' and ...
). Examples of Bernoulli trials include: *Flipping a coin. In this context, obverse ("heads") conventionally denotes success and reverse ("tails") denotes failure. A
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. In the ...
has the probability of success 0.5 by definition. In this case there are exactly two possible outcomes. *Rolling a , 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 An opinion poll, often simply referred to as a survey or a poll (although strictly a poll is an actual election) is a human research survey of public opinion from a particular sample. Opinion polls are usually designed to represent the opinions ...
, 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 exactly two possible outcomes are called Bernoulli trials. Call one of the outcomes "success" and the other outcome "failure". Let p be the probability of success in a Bernoulli trial, and q 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 In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
and exhaustive. Thus one has the following relations: : p = 1 - q, \quad \quad q = 1 - p, \quad \quad p + q = 1. Alternatively, these can be stated in terms of
odds Odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics. Odds also have ...
: given probability ''p'' of success and ''q'' of failure, the ''odds for'' are p:q and the ''odds against'' are q:p. These can also be expressed as numbers, by dividing, yielding the odds for, o_f, and the odds against, o_a:, : \begin o_f &= p/q = p/(1-p) = (1-q)/q\\ o_a &= q/p = (1-p)/p = q/(1-q) \end These are
multiplicative inverse In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rat ...
s, so they multiply to 1, with the following relations: : o_f = 1/o_a, \quad o_a = 1/o_f, \quad o_f \cdot o_a = 1. In the case that a Bernoulli trial is representing an event from finitely many
equally likely outcomes In probability theory, an outcome is a possible result of an experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experimen ...
, where ''S'' of the outcomes are success and ''F'' of the outcomes are failure, the odds for are S:F and the odds against are F:S. This yields the following formulas for probability and odds: : \begin p &= S/(S+F)\\ q &= F/(S+F)\\ o_f &= S/F\\ o_a &= F/S \end 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 variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s 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 n of
statistically independent Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes. Two events are independent, statistically independent, or stochastically independent if, informally speaking, the occurrence of o ...
Bernoulli trials, each with a probability of success p, and counts the number of successes. A random variable corresponding to a binomial experiment is denoted by B(n,p), and is said to have a '' binomial distribution''. The probability of exactly k successes in the experiment B(n,p) is given by: :P(k)= p^k q^ where is a
binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
. 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 trial In survey methodology, Poisson sampling (sometimes denoted as ''PO sampling'') is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sampl ...
s.
Rajeev Motwani Rajeev Motwani (Hindi: राजीव मोटवानी , March 24, 1962 – June 5, 2009) was an Indian American professor of Computer Science at Stanford University whose research focused on theoretical computer science. He was an early ad ...
and P. Raghavan. Randomized Algorithms. Cambridge University Press, New York (NY), 1995, p.67-68


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 p = \tfrac. Thus the probability of failure, q, is given by :q = 1 - p = 1 - \tfrac = \tfrac. Using the equation above, the probability of exactly two tosses out of four total tosses resulting in a heads is given by: :\begin P(2) &= p^ q^ \\ &= 6 \times \left(\tfrac\right)^2 \times \left(\tfrac\right)^2 \\ &= \dfrac . \end


See also

*
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 appear naturally in symbolic dynamics, and are thus important in the study of dynamical sy ...
*
Bernoulli sampling In the theory of finite population sampling, Bernoulli sampling is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample. An essential p ...
*
Bernoulli distribution In probability theory and statistics, the Bernoulli distribution, named after Swiss mathematician Jacob Bernoulli,James Victor Uspensky: ''Introduction to Mathematical Probability'', McGraw-Hill, New York 1937, page 45 is the discrete probabil ...
* Binomial distribution *
Binomial coefficient In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
*
Binomial proportion confidence interval In statistics, a binomial proportion confidence interval is a confidence interval for the probability of success calculated from the outcome of a series of success–failure experiments (Bernoulli trial, Bernoulli trials). In other words, a binomia ...
*
Poisson sampling In survey methodology, Poisson sampling (sometimes denoted as ''PO sampling'') is a sampling process where each element of the population is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sampl ...
*
Sampling design Sampling may refer to: *Sampling (signal processing), converting a continuous signal into a discrete signal * Sampling (graphics), converting continuous colors into discrete color components *Sampling (music), the reuse of a sound recording in ano ...
*
Coin flipping Coin flipping, coin tossing, or heads or tails is the practice of throwing a coin in the air and checking which side is showing when it lands, in order to choose between two alternatives, heads or tails, sometimes used to resolve a dispute betwe ...
*
Jacob Bernoulli Jacob Bernoulli (also known as James or Jacques; – 16 August 1705) was one of the many prominent mathematicians in the Bernoulli family. He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during the Le ...
*
Fisher's exact test Fisher's exact test is a statistical significance test used in the analysis of contingency tables. Although in practice it is employed when sample sizes are small, it is valid for all sample sizes. It is named after its inventor, Ronald Fisher, a ...
*
Boschloo's test Boschloo's test is a statistical hypothesis test for analysing 2x2 contingency tables. It examines the association of two Bernoulli distributed random variables and is a uniformly more powerful alternative to Fisher's exact test. It was propo ...


References


External links

* * {{DEFAULTSORT:Bernoulli Trial Discrete distributions Coin flipping Experiment (probability theory)