{{Short description|Event that contains only one outcome}} {{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 which contains only a single 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. 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 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.<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-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 <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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