# Elementary event

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 corresponds to precisely one outcome.

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The following are examples of elementary events:

- All sets {
*k*}, where*k*∈**N**if objects are being counted and the sample space is*S*= {0, 1, 2, 3, ...} (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. - All sets {
*x*}, where*x*is a real number. Here*X*is a random variable with a normal distribution and*S*= (−∞, +∞). 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 because there are infinitely many of them— then non-zero probabilities can only be assigned to non-elementary events.

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.

## See also

## References

- Wackerly, Denniss; William Mendenhall; Richard Scheaffer.
*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.

## Further reading

- 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.