elementary event

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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.{{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}} 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 $k \in \N$ 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.{{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}}

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.

elementary event

Probability of an elementary event

See also

References

Further reading