# 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 corresponding to precisely one outcome.

The following are examples of elementary events:

• All sets $\displaystyle{ \{ k \}, }$ where $\displaystyle{ k \in \N }$ if objects are being counted and the sample space is $\displaystyle{ S = \{ 1, 2, 3, \ldots \} }$ (the natural numbers).
• $\displaystyle{ \{ HH \}, \{ HT \}, \{ TH \}, \text{ and } \{ TT \} }$ if a coin is tossed twice. $\displaystyle{ S = \{ HH, HT, TH, TT \} }$ where $\displaystyle{ H }$ stands for heads and $\displaystyle{ T }$ for tails.
• All sets $\displaystyle{ \{ x \}, }$ where $\displaystyle{ x }$ is a real number. Here $\displaystyle{ X }$ is a random variable with a normal distribution and $\displaystyle{ S = (-\infty, + \infty). }$ 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 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 $\displaystyle{ S }$ and not necessarily the full power set.

## References

1. Wackerly, Denniss; William Mendenhall; Richard Scheaffer. Mathematical Statistics with Applications. Duxbury. ISBN 0-534-37741-6.
2. 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.