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.^[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 ${\displaystyle k\in \mathbb {N} }$ 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.

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 ${\displaystyle S}$ 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

• 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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