Editing Event (probability theory) (section)

{{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).