Editing Elementary event

{{redirect2|Basic outcome|Atomic event|atomic events in computer science|linearizability}}

{{Probability fundamentals}}
In [[probability theory]], an '''elementary event''', also called an '''atomic event''' or '''sample point''', is an [[Event (probability theory)|event]] which contains only a single [[Outcome (probability)|outcome]] in the [[sample space]].<ref>{{cite book|last=Wackerly|first=Denniss|author2=William Mendenhall|author3=Richard Scheaffer|title=Mathematical Statistics with Applications|year=2002 |publisher=Duxbury|isbn=0-534-37741-6}}</ref> Using [[set theory]] terminology, an elementary event is a [[Singleton (mathematics)|singleton]]. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponding to precisely one outcome.

The following are examples of elementary events:
* All sets <math>\{ k \},</math> where <math>k \in \N</math> if objects are being counted and the sample space is <math>S = \{ 1, 2, 3, \ldots \}</math> (the [[natural numbers]]).
* <math>\{ HH \}, \{ HT \}, \{ TH \}, \text{ and } \{ TT \}</math> if a coin is tossed twice. <math>S = \{ HH, HT, TH, TT \}</math> where <math>H</math> stands for heads and <math>T</math> for tails.
* All sets <math>\{ x \},</math> where <math>x</math> is a  [[real number]]. Here <math>X</math> is a [[random variable]] with a [[normal distribution]] and <math>S = (-\infty, + \infty).</math> This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous [[probability distribution]]..

==Probability of an elementary event==

Elementary events may occur with probabilities that are between zero and one (inclusively). In a [[Discrete random variable|discrete]] probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in a [[Continuous random variable|continuous]] distribution, individual elementary events must all have a probability of zero.

Some "mixed" distributions contain both stretches of continuous elementary events and some discrete elementary events; the discrete elementary events in such distributions can be called '''atoms''' or '''atomic events''' and can have non-zero probabilities.<ref>{{cite book|last=Kallenberg|first=Olav|title=Foundations of Modern Probability|edition=2nd|year=2002|page=9|url=https://books.google.com/books?id=L6fhXh13OyMC|publisher=Springer|location=New York|isbn=0-387-94957-7}}</ref>

Under the [[Measure theory|measure-theoretic]] definition of a [[probability space]], the probability of an elementary event need not even be defined. In particular, the set of events on which probability is defined may be some [[Sigma-algebra|σ-algebra]] on <math>S</math> and not necessarily the full [[power set]].

==See also==

* {{annotated link|Atom (measure theory)}}
* {{annotated link|Pairwise independence|Pairwise independent events}}

==References==

{{reflist}}

==Further reading==

* {{cite book|last=Pfeiffer|first=Paul E.|year=1978|title=Concepts of Probability Theory|publisher=Dover|isbn=0-486-63677-1|page=18}}
* {{cite book|last=Ramanathan|first=Ramu|title=Statistical Methods in Econometrics|location=San Diego|publisher=Academic Press|year=1993|isbn=0-12-576830-3|pages=7–9}}

[[Category:Experiment (probability theory)]]


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