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Event (probability theory)

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In statistics and probability theory, set of outcomes to which a probability is assigned

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Probability theory
Part of a series onstatistics

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
Venn diagram Tree diagram
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Inprobability theory, anevent is asubset ofoutcomes of anexperiment (asubset of thesample space) to which a probability is assigned.^[1] A single outcome may be an element of many different events,^[2] and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.^[3] An event consisting of only a single outcome is called anelementary event or anatomic event; that is, it is asingleton set. An event that has more than one possible outcome is called acompound event. An event $S {\displaystyle S}$ is said tooccur if $S {\displaystyle S}$ contains the outcome $x {\displaystyle x}$ of theexperiment (or trial) (that is, if $x\in S$ ).^[4] The probability (with respect to someprobability measure) that an event $S {\displaystyle S}$ occurs is the probability that $S {\displaystyle S}$ contains the outcome $x {\displaystyle x}$ of an experiment (that is, it is the probability that $x\in S$ ). An event defines acomplementary event, namely the complementary set (the eventnot occurring), and together these define aBernoulli trial: did the event occur or not?

Typically, when thesample space is finite, any subset of the sample space is an event (that is, all elements of thepower set of the sample space are defined as events).^[5] However, this approach does not work well in cases where the sample space isuncountably infinite. So, when defining aprobability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see§ Events in probability spaces, below).

A simple example

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If we assemble a deck of 52playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including anysingleton set (anelementary event), theempty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events areproper subsets of the sample space that contain multiple elements. So, for example, potential events include:

AnEuler diagram of an event. $B {\displaystyle B}$ is the sample space and $A {\displaystyle A}$ is an event.
By the ratio of their areas, the probability of $A {\displaystyle A}$ is approximately 0.4.

AnEuler diagram of an event. $B {\displaystyle B}$ is the sample space and $A {\displaystyle A}$ is an event.
By the ratio of their areas, the probability of $A {\displaystyle A}$ is approximately 0.4.

"Red and black at the same time without being a joker" (0 elements),
"The 5 of Hearts" (1 element),
"A King" (4 elements),
"A Face card" (12 elements),
"A Spade" (13 elements),
"A Face card or a red suit" (32 elements),
"A card" (52 elements).

Since all events are sets, they are usually written as sets (for example, {1, 2, 3}), and represented graphically usingVenn diagrams. In the situation where each outcome in the sample space Ω is equally likely, the probability $P {\displaystyle P}$ of an event $A {\displaystyle A}$ is the followingformula: $\mathrm {P} (A)={\frac {|A|}{|\Omega |}}\,\ \left({\text{alternatively:}}\ \Pr(A)={\frac {|A|}{|\Omega |}}\right)$ This rule can readily be applied to each of the example events above.

Events in probability spaces

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Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standardprobability distributions, such as thenormal distribution, the sample space is the set of real numbers or some subset of thereal numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers'badly behaved' sets, such as those that arenonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such asjoint andconditional probabilities, to work, it is necessary to use aσ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice ofσ-algebra is theBorel measurable set derived from unions and intersections of intervals. However, the larger class ofLebesgue measurable sets proves more useful in practice.

In the generalmeasure-theoretic description ofprobability spaces, an event may be defined as an element of a selected𝜎-algebra of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the 𝜎-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, allevents of interest are elements of the 𝜎-algebra.