Elementary event

"Basic outcome" and "Atomic event" redirect here. For atomic events in computer science, see linearizability.

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..

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.

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.

See also

• 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

References

1. Wackerly, Denniss; William Mendenhall; Richard Scheaffer (2002). 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.