• Source: 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. 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



      {
      k
      }
      ,


      {\displaystyle \{k\},}

      where



      k
      ∈

      N



      {\displaystyle k\in \mathbb {N} }

      if objects are being counted and the sample space is



      S
      =
      {
      1
      ,
      2
      ,
      3
      ,
      …
      }


      {\displaystyle S=\{1,2,3,\ldots \}}

      (the natural numbers).




      {
      H
      H
      }
      ,
      {
      H
      T
      }
      ,
      {
      T
      H
      }
      ,

      and

      {
      T
      T
      }


      {\displaystyle \{HH\},\{HT\},\{TH\},{\text{ and }}\{TT\}}

      if a coin is tossed twice.



      S
      =
      {
      H
      H
      ,
      H
      T
      ,
      T
      H
      ,
      T
      T
      }


      {\displaystyle S=\{HH,HT,TH,TT\}}

      where



      H


      {\displaystyle H}

      stands for heads and



      T


      {\displaystyle T}

      for tails.
      All sets



      {
      x
      }
      ,


      {\displaystyle \{x\},}

      where



      x


      {\displaystyle x}

      is a real number. Here



      X


      {\displaystyle X}

      is a random variable with a normal distribution and



      S
      =
      (
      −
      ∞
      ,
      +
      ∞
      )
      .


      {\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.
      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


      {\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




      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.

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