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Experiment (probability theory)

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Procedure that can be infinitely repeated, with a well-defined set of outcomes

This article is about the probabilistic model used in actual experiments. For a discussion about actual experiments, seeexperiment.

Probability theory
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Probability Axioms Determinism System Indeterminism Randomness
Probability space Sample space Event Collectively exhaustive events Elementary event Mutual exclusivity Outcome Singleton Experiment Bernoulli trial Probability distribution Bernoulli distribution Binomial distribution Exponential distribution Normal distribution Pareto distribution Poisson distribution Probability measure Random variable Bernoulli process Continuous or discrete Expected value Variance Markov chain Observed value Random walk Stochastic process
Complementary event Joint probability Marginal probability Conditional probability
Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality
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Inprobability theory, anexperiment ortrial (see below) is themathematical model of anyprocedure that can beinfinitely repeated and has awell-defined set of possibleoutcomes, known as thesample space.^[1] An experiment is said to berandom if it has more than one possible outcome, anddeterministic if it has only one. A random experiment that has exactly two (mutually exclusive) possible outcomes is known as aBernoulli trial.^[2]

When an experiment is conducted, one (and only one) outcome results— although this outcome may be included in any number ofevents, all of which would be said to have occurred on that trial. After conducting many trials of the same experiment and pooling the results, an experimenter can begin to assess theempirical probabilities of the various outcomes and events that can occur in the experiment and apply the methods ofstatistical analysis.

Experiments and trials

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Random experiments are often conducted repeatedly, so that the collective results may be subjected tostatistical analysis. A fixed number of repetitions of the same experiment can be thought of as acomposed experiment, in which case the individual repetitions are calledtrials. For example, if one were to toss the same coin one hundred times and record each result, each toss would be considered a trial within the experiment composed of all hundred tosses.^[3]

Mathematical description

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Main article:Probability space

A random experiment is described or modeled by a mathematical construct known as aprobability space. A probability space is constructed and defined with a specific kind of experiment or trial in mind.

A mathematical description of an experiment consists of three parts:

Asample space, Ω (orS), which is theset of all possibleoutcomes.
A set ofevents $\scriptstyle {\mathcal {F}}$ , where each event is a set containing zero or more outcomes.
The assignment ofprobabilities to the events—that is, a functionP mapping from events to probabilities.

Anoutcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complicatedevents are used to characterize groups of outcomes. The collection of all such events is asigma-algebra $\scriptstyle {\mathcal {F}}$ . Finally, there is a need to specify each event's likelihood of happening; this is done using theprobability measure function,P.

Once an experiment is designed and established,ω from the sample space Ω, all the events in $\scriptstyle {\mathcal {F}}$ that contain the selected outcomeω (recall that each event is a subset of Ω) are said to “have occurred”. The probability functionP is defined in such a way that, if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events wouldapproach agreement with the valuesP assigns them.

As a simple experiment, we may flip a coin twice. The sample space (where the order of the two flips is relevant) is{(H, T), (T, H), (T, T), (H, H)} where "H" means "heads" and "T" means "tails". Note that each of(H, T), (T, H), ... are possibleoutcomes of the experiment. We may define anevent which occurs when a "heads" occurs in either of the two flips. This event contains all of the outcomes except(T, T).