Elementary Event

In probability theory, an elementary event (also called an atomic event or simple event) is an event which contains only a single outcome in the sample space. Set theoretically this means that an elementary event is a singleton. Elementary events and the corresponding outcome are often written interchangeably for simplicity, as such an event corresponds to precisely one outcome.

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. S = {HH, HT, TH, TT}. 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 = (−∞, +∞). This example shows that, because the probability of each elementary event is 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 whose sample space is finite is determined by the probabilities it assigns to 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. 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.