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In probability theory, an elementary event or atomic event is
a subset of a sample space that contains only one element of the sample space. It is important to note that an elementary event is still a
set containing an element of the sample space, not that element itself. However, elementary events are often written as
elements rather than sets for simplicity, where this is unambiguous.
Examples of sample spaces, S, and elementary events include:
- If objects are being counted, and the sample space S = {0, 1, 2, 3, ...} (the natural numbers), then the elementary events are all sets {k}, where k ∈
N.
- If a coin is tossed twice, S = {HH, HT, TH, TT}, H for heads and T for tails, and the elementary events are {HH},
{HT}, {TH} and {TT}.
- If X is a Gaussian random variable, S =
(-∞, +∞), the real numbers, and the elementary events are all
sets {x}, where x ∈ R.
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, which 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.
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