Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p. Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to 0. Grey curve: To get 50-50 chance of throwing a Yahtzee (5 cubic dice all showing the same number) requires 0.69 × 1296 ~ 898 throws. Green curve: Drawing a card from a deck of playing cards without jokers 100 (1.92 × 52) times with replacement gives 85.7% chance of drawing the ace of spades at least once.
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.
^Papoulis, A. (1984). "Bernoulli Trials". Probability, Random Variables, and Stochastic Processes (2nd ed.). New York: McGraw-Hill. pp. 57–63.
^James Victor Uspensky: Introduction to Mathematical Probability, McGraw-Hill, New York 1937, page 45
In the theory of probability and statistics, a Bernoullitrial (or binomial trial) is a random experiment with exactly two possible outcomes, "success"...
{\displaystyle p\neq 1/2.} The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted (so n would be 1...
associated with a Bernoullitrial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also...
Jacob Bernoulli (also known as James in English or Jacques in French; 6 January 1655 [O.S. 27 December 1654] – 16 August 1705) was one of the many prominent...
is also called a Bernoullitrial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the...
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distributed values Basic distributions: Bernoulli distribution, for the outcome of a single Bernoullitrial (e.g. success/failure, yes/no) Binomial distribution...
{\displaystyle \mathrm {H} (X)} , the entropy function. Mathematically, the Bernoullitrial is modelled as a random variable X {\displaystyle X} that can take...
A. Given an event, the event and its complementary event define a Bernoullitrial: did the event occur or not? For example, if a typical coin is tossed...
population sampling, Bernoulli sampling is a sampling process where each element of the population is subjected to an independent Bernoullitrial which determines...
calculated from the outcome of a series of success–failure experiments (Bernoullitrials). In other words, a binomial proportion confidence interval is an interval...
possible outcomes of n independent Bernoullitrials and x i j {\displaystyle x_{ij}} is the j th outcome of the i th trial, then the probability of x i {\displaystyle...
whom the Bernoulli brothers had recommended. His early death cut short a promising career. Bernoulli distribution Bernoulli process Bernoullitrial St. Petersburg...
Bernoulli number Bernoulli polynomials Bernoulli process Bernoullitrial Lemniscate of BernoulliBernoulli, a journal published by the Bernoulli Society for...
where each element of the population is subjected to an independent Bernoullitrial which determines whether the element becomes part of the sample.: 85 ...
and second coin flips respectively. Each coin flip is a Bernoullitrial and has a Bernoulli distribution. If a coin displays "heads" then the associated...
probability of success in a series of Bernoullitrials will converge to the theoretical probability. For a Bernoulli random variable, the expected value...
interval can be seen as a sequence of n Bernoullitrials, where the i {\displaystyle i} -th Bernoullitrial corresponds to looking whether an event happens...
special interest for use with scale parameters. As a concrete example, a Bernoulli distribution can be parametrized by the probability of occurrence p, or...
determines the suffix, and k the prefix). The Bernoulli distribution models the outcome of a single Bernoullitrial. In other words, it models whether flipping...
yearly Bernoullitrial in the binomial distribution. That is disfavoured because each year does not represent an independent Bernoullitrial but is an...
event Mutual exclusivity Outcome Singleton Experiment Bernoullitrial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution...
being independent of the past states). A Bernoulli scheme with only two possible states is known as a Bernoulli process. Note, however, by the Ornstein...
event Mutual exclusivity Outcome Singleton Experiment Bernoullitrial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution...
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