Elementary event
# 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 $k\in \mathbb {N}$ 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.

