# Elementary event

## From Wikipedia, the free encyclopedia

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.

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.

- Atom (measure theory) – A measurable set with positive measure that contains no subset of smaller positive measure
- Pairwise independent events – Set of random variables of which any two are independent

- 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.

- Pfeiffer, Paul E. (1978).
*Concepts of Probability Theory*. Dover. p. 18. ISBN 0-486-63677-1. - Ramanathan, Ramu (1993).
*Statistical Methods in Econometrics*. San Diego: Academic Press. pp. 7–9. ISBN 0-12-576830-3.

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