Elementary event
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.