# Elementary event

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In probability theory, an elementary event or atomic event is a singleton of a sample space. An outcome is an element of a sample space. An elementary event is a set containing an outcome, not the outcome itself. However, elementary events are often written as outcomes for simplicity when the difference is unambiguous.

The following are examples of elementary events:

• All sets {k}, where k ∈ N if objects are being counted and the sample space is S = {0, 1, 2, 3, ...} (the natural numbers).
• {HH}, {HT}, {TH} and {TT} if a coin is tossed twice[dubious ]. S = {HH, HT, TH, TT}. H stands for heads and T for tails.
• The real numbers, and the elementary events, are all sets {x}, where x ∈ R if X is a normally distributed random variable and S = (−∞, +∞). This example shows that, because they are all zero, the probabilities assigned to atomic events do not determine a continuous probability distribution.

Elementary events may have probabilities that are strictly positive, zero, undefined, or any combination thereof. For instance, any discrete probability distribution is determined by the probabilities it assigns to what may be called elementary events. In contrast, all elementary events have probability zero under any continuous distribution. Mixed distributions, being neither entirely continuous nor entirely discrete, may contain atoms. Atoms can be thought of as elementary (that is, atomic) events with non-zero probabilities. Under the measure-theoretic definition of a probability space, the probability of an elementary event need not even be defined, since mathematicians distinguish between the sample space S and the events of interest, defined by the elements of a σ-algebra on S.