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Inprobability theory, anelementary event, also called anatomic event orsample point, is anevent which contains only a singleoutcome in thesample space.^[1] Usingset theory terminology, an elementary event is asingleton. 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${\displaystyle S=\{1,2,3,\dots \}}$ (thenatural 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 areal number. Here${\displaystyle X}$ is arandom variable with anormal distribution and${\displaystyle S=(-\mathrm{\infty },+\mathrm{\infty }).}$ This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuousprobability distribution..

Elementary events may occur with probabilities that are between zero and one (inclusively). In adiscrete probability distribution whose sample space is finite, each elementary event is assigned a particular probability. In contrast, in acontinuous 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 calledatoms oratomic events and can have non-zero probabilities.^[2]

Under themeasure-theoretic definition of aprobability 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 someσ-algebra on${\displaystyle S}$ and not necessarily the fullpower 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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