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Event (probability theory)

In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. Typically, any subset of the sample space is an event (i.e. all elements of the power set of the sample space are events), but when defining a probability space it is possible to exclude certain subsets of the sample space from being events (see §2, below).

A simple example

If we assemble a deck of 52 playing cards and two jokers, and draw a single card from the deck, then the sample space is a 54-element set, at each individual card is a possible outcome. An event, however, is any subset of the sample space, including any single-element set (an elementary event, of which there are 54, representing the 54 possible cards drawn from the deck), the empty set (which is defined to have probability zero) and the entire set of 54 cards, the sample space itself (which is defined to have probability one). Other events are sets are proper subsets of the sample space that contain multiple elements, for example:

Since all events are sets, they are usually written as sets (e.g. {1, 2, 3}), and represented graphically using Venn diagrams. Venn diagrams are particularly useful for representing events because the probability of the event can be identified with the ratio of the area of the event and the area of the sample space. (Indeed, each of the axioms of probability, and the definition of conditional probability can be represented in this fashion.)

A Venn diagram of an event. A is the sample space and B is an event.
By the ratio of their areas, the probability of B is approximately 0.3.

Events in probability spaces

In the measure-theoretic description of probability spaces, an event may be defined as an element of the σ-algebra on the sample space. Note, however, that under this definition, any subset of the sample space that is not an element of the σ-algebra is not technically an event, and does not have a probability. Any confusion may be resolved by considering any subset of the sample space to be an event, but only the elements of the σ-algebra to be events of interest.