View source for Event (probability theory)

{{Short description|In statistics and probability theory, set of outcomes to which a probability is assigned}}
{{refimprove|date=January 2018}}

{{Probability fundamentals}}
In [[probability theory]], an '''event''' is a [[subset]] of [[Outcome (probability)|outcomes]] of an [[Experiment (probability theory)|experiment]] (a [[subset]] of the [[sample space]]) to which a probability is assigned.<ref>{{cite book|last=Leon-Garcia|first=Alberto|title=Probability, statistics and random processes for electrical engineering|location=Upper Saddle River, NJ|publisher=Pearson|year=2008|isbn=9780131471221 |url=https://books.google.com/books?id=GUJosCkbBywC}}</ref> A single outcome may be an element of many different events,<ref>{{cite book|last=Pfeiffer|first=Paul E.|year=1978|title=Concepts of probability theory|page=18|url=https://books.google.com/books?id=_mayRBczVRwC&pg=PA18|publisher=Dover Publications|isbn=978-0-486-63677-1}}</ref> and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.<ref>{{cite book|last=Foerster|first=Paul A.|year=2006|title=Algebra and trigonometry: Functions and Applications, Teacher's edition|edition=Classics|page=[https://archive.org/details/algebratrigonome00paul_0/page/634 634]|publisher=[[Prentice Hall]]|location=Upper Saddle River, NJ|isbn=0-13-165711-9|url=https://archive.org/details/algebratrigonome00paul_0/page/634 }}</ref> An event consisting of only a single outcome is called an {{em|[[elementary event]]}} or an {{em|atomic event}}; that is, it is a [[singleton set]]. An event that has more than one possible outcome is called a '''compound event.''' An event <math>S</math> is said to {{em|occur}} if <math>S</math> contains the outcome <math>x</math> of the [[Experiment (probability theory)|experiment]] (or trial) (that is, if <math>x \in S</math>).<ref>{{Cite book |last1=Dekking |first1=Frederik Michel |url=https://link.springer.com/book/10.1007/1-84628-168-7 |title=A modern introduction to probability and statistics: understanding why and how |last2=Kraaikamp |first2=Cornelis |last3=Lopuhaä |first3=Hendrik Paul |last4=Ludolf Erwin |first4=Meester |date=2005 |publisher=Springer |isbn=978-1-85233-896-1 |editor-last=Dekking |editor-first=Michel |series=Springer texts in statistics |location=London [Heidelberg] |pages=14|doi=10.1007/1-84628-168-7 }}</ref> The probability (with respect to some [[probability measure]]) that an event <math>S</math> occurs is the probability that <math>S</math> contains the outcome <math>x</math> of an experiment (that is, it is the probability that <math>x \in S</math>). 
An event defines a [[complementary event]], namely the complementary set (the event {{em|not}} occurring), and together these define a [[Bernoulli trial]]: did the event occur or not?

Typically, when the [[sample space]] is finite, any subset of the sample space is an event (that is, all elements of the [[power set]] of the sample space are defined as events).<ref>{{Cite book |last=Širjaev |first=Alʹbert N. |title=Probability-1 |date=2016 |publisher=Springer |isbn=978-0-387-72205-4 |edition=3rd |series=Graduate texts in mathematics |location=New York Heidelberg Dordrecht London |translator-last=Boas |translator-first=Ralph Philip |translator-last2=Chibisov |translator-first2=Dmitry}}</ref> However, this approach does not work well in cases where the sample space is [[uncountably infinite]]. So, when defining a [[probability space]] it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see {{section link||Events in probability spaces}}, below).

==A simple example==

If we assemble a deck of 52 [[playing card]]s 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 any [[singleton set]] (an [[elementary event]]), the [[empty set]] (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one).  Other events are [[proper subset]]s of the sample space that contain multiple elements.  So, for example, potential events include:  
[[Image:Venn A subset B.svg|thumb|150px|An [[Euler diagram]] of an event. <math>B</math> is the sample space and <math>A</math> is an event.<br>By the ratio of their areas, the probability of <math>A</math> 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 using [[Venn diagram]]s. In the situation where each outcome in the sample space Ω is equally likely, the probability <math>P</math> of an event <math>A</math> is the following {{visible anchor|formula}}:
<math display=block>\mathrm{P}(A) = \frac{|A|}{|\Omega|}\,\ \left( \text{alternatively:}\ \Pr(A) = \frac{|A|}{|\Omega|}\right)</math>
This rule can readily be applied to each of the example events above.

==Events in probability spaces==<!--Section linked from lead-->

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 standard [[probability distributions]], such as the [[normal distribution]], the sample space is the set of real numbers or some subset of the [[real numbers]]. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers [[Pathological (mathematics)|'badly behaved']] sets, such as those that are [[Nonmeasurable set|nonmeasurable]]. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as [[Joint probability|joint]] and [[conditional probability|conditional probabilities]], to work, it is necessary to use a [[sigma-algebra|&sigma;-algebra]], that is, a family closed under complementation and countable unions of its members. The most natural choice of [[Sigma-algebra|&sigma;-algebra]] is the [[Borel measure|Borel measurable]] set derived from unions and intersections of intervals. However, the larger class of [[Lebesgue measure|Lebesgue measurable]] sets proves more useful in practice.

In the general [[Measure theory|measure-theoretic]] description of [[probability space]]s, an event may be defined as an element of a selected [[sigma-algebra|{{sigma}}-algebra]] of subsets of the sample space. Under this definition, any subset of the sample space that is not an element of the {{sigma}}-algebra is not an event, and does not have a probability.  With a reasonable specification of the probability space, however, all {{em|events of interest}} are elements of the {{sigma}}-algebra.

==A note on notation==

Even though events are subsets of some sample space <math>\Omega,</math> they are often written as predicates or indicators involving [[random variable]]s. For example, if <math>X</math> is a real-valued random variable defined on the sample space <math>\Omega,</math> the event
<math display=block>\{ \omega \in \Omega \mid u < X(\omega) \leq v \}\,</math>
can be written more conveniently as, simply,
<math display=block>u < X \leq v\,.</math>
This is especially common in formulas for a [[probability]], such as
<math display=block>\Pr(u < X \leq v) = F(v) - F(u)\,.</math>
The [[Set (mathematics)|set]] <math>u < X \leq v</math> is an example of an [[inverse image]] under the [[Map (mathematics)|mapping]] <math>X</math> because <math>\omega \in X^{-1}((u, v])</math> if and only if <math>u < X(\omega) \leq v.</math>

==See also==

* {{annotated link|Atom (measure theory)}}
* {{annotated link|Complementary event}}
* {{annotated link|Elementary event}}
* {{annotated link|Independent event}}
* {{annotated link|Outcome (probability)}}
* {{annotated link|Pairwise independence|Pairwise independent events}}

==Notes==

{{reflist}}

==External links==

{{Commonscat}}
* {{springer|title=Random event|id=p/r077290}}
* [https://web.archive.org/web/20130923121802/http://mws.cs.ru.nl/mwiki/prob_1.html#M1 Formal definition] in the [[Mizar system]].

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[[Category:Experiment (probability theory)]]