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 $$\{ k \},$$ where $$k \in \N$$ if objects are being counted and the sample space is $$S = \{ 1, 2, 3, \ldots \}$$ (the natural numbers).
 * $$\{ HH \}, \{ HT \}, \{ TH \}, \text{ and } \{ TT \}$$ if a coin is tossed twice. $$S = \{ HH, HT, TH, TT \}$$ where $$H$$ stands for heads and $$T$$ for tails.
 * All sets $$\{ x \},$$ where $$x$$ is a real number. Here $$X$$ is a random variable with a normal distribution and $$S = (-\infty, + \infty).$$ 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 $$S$$ and not necessarily the full power set.