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In probability theory, an elementary event, also called an atomic event or sample point, is an event which contains only a single outcome in the sample space.^[1] Using set theory terminology, an elementary event is a 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 $\{k\},$ where $k\in \mathbb {N}$ if objects are being counted and the sample space is $S=\{1,2,3,\ldots \}$ (the natural numbers).
$\{HH\},\{HT\},\{TH\},{\text{ and }}\{TT\}$ if a coin is tossed twice. $S=\{HH,HT,TH,TT\}$ where $H$ stands for heads and $T$ for tails.
All sets $\{x\},$ where $x$ is a real number. Here $X$ is a random variable with a normal distribution and $S=(-\infty ,+\infty ).$ 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 probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in a 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.^[2]

Under the 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 σ-algebra on $S$ and not necessarily the full power set.

References

^ Wackerly, Denniss; William Mendenhall; Richard Scheaffer (2002). Mathematical Statistics with Applications. Duxbury. ISBN 0-534-37741-6.
^ Kallenberg, Olav (2002). Foundations of Modern Probability (2nd ed.). New York: Springer. p. 9. ISBN 0-387-94957-7.

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-{{redirect2|Basic outcome | Atomic event|atomic events in computer science|linearizability}}
+{{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 | 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 corresponds to precisely one outcome.
+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 {''k''}, where ''k''&nbsp;∈&nbsp;'''N''' if objects are being counted and the sample space is ''S'' = {0, 1, 2, 3, ...} (the [[natural numbers]]).
+* 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]]).
-* {HH}, {HT}, {TH} and {TT} if a coin is tossed twice. ''S'' = {HH, HT, TH, TT}. H stands for heads and T for tails.
+* <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 {''x''}, where ''x'' is a  [[real number]]. Here ''X'' is a [[random variable]] with a [[normal distribution]] and ''S''&nbsp;=&nbsp;(&minus;∞,&nbsp;+∞).  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]].
+* 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 because there are infinitely many of them&mdash; then non-zero probabilities can only be assigned to non-elementary events.
+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/about/Foundations_of_Modern_Probability.html?id=L6fhXh13OyMC | publisher = Springer | location = New York | isbn = 0-387-94957-7}}</ref>
+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 ''S'' and not necessarily the full [[power set]].
+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==
-*[[Atom (measure theory)]]
+* {{annotated link|Atom (measure theory)}}
-*[[Pairwise independence|Pairwise independent events]]
+* {{annotated link|Pairwise independence|Pairwise independent events}}
 ==References==
-{{Reflist}}
+{{reflist}}
 ==Further reading==
-*{{cite book |last=Pfeiffer |first=Paul E. |year=1978 |title=Concepts of Probability Theory |location= |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 }}
+* {{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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 {{probability-stub}}
-{{statistics-stub}}