In
probability theory, an
event is a
set of outcomes (a
subset of the
sample space) to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event (
i.
e. all elements of the
power set of the sample space are defined as events). However, this approach does not work well in cases where the sample space is infinite, most notably when the outcome is a real number. So, when defining a
probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see §2, below).
If we assemble a deck of 52 playing cards and no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as 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 52, representing the 52 possible cards drawn from the deck), the empty set (which is defined to have probability zero) and the entire set of 52 cards, the sample space itself (which is defined to have probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include
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.)
Defining all subsets of the sample space as events works well when there are only finitely many outcomes, but gives rise to problems when the sample space is infinite. For many standard probability distributions, such as the normal distribution the sample space is the set of real numbers or some subset of the real numbers. Attempts to define probabilities for all subsets of the real numbers run into difficulties when one considers 'badly-behaved' sets, such as those which are nonmeasurable. Hence, it is necessary to restrict attention to a more limited family of subsets. For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a s-algebra, that is, a family closed under countable unions and intersections. The most natural choice is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice.
