In probability theory, an event is a set of outcomes (a subset of the sample space) to which a probability is assigned. Probability theory is the branch of Mathematics concerned with analysis of random phenomena In Probability theory, the sample space or universal sample space, often denoted S, Ω or U (for "universe" of an Experiment 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). In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) 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). A probability space, in Probability theory, is the conventional Mathematical model of Randomness.

A simple example

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. A playing card is a piece of specially prepared heavy paper thin card or thin plastic figured with distinguishing motifs and used as one of a set for playing Card games 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). In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the In Probability theory, an elementary event or atomic event is a subset of a Sample space that contains only one element In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:

A Venn diagram of an event. Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups B is the sample space and A is an event.
By the ratio of their areas, the probability of A is approximately 0. 4.

• "Red and black at the same time without being a joker" (0 elements),
• "The 5 of Hearts" (1 element),
• "A King" (4 elements),
• "A Face card" (12 elements),
• "A Spade" (13 elements),
• "A Face card or a red suit" (32 elements),
• "A card" (52 elements).

Since all events are sets, they are usually written as sets (e. g. {1, 2, 3}), and represented graphically using Venn diagrams. Venn diagrams or set diagrams are Diagrams that show all hypothetically possible Logical relations between a finite collection of sets (groups 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. In Probability theory, the Probability P of some event E, denoted P(E is defined in such a way that P satisfies the Conditional probability is the Probability of some event A, given the occurrence of some other event B. )

Events in probability spaces

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. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable The normal distribution, also called the Gaussian distribution, is an important family of Continuous probability distributions applicable in many fields In Mathematics, the real numbers may be described informally in several different ways 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. This page gives a non-technical description of this concept For a technical description see Measure (mathematics and the various constructions of non-measurable sets 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 σ-algebra, that is, a family closed under countable unions and intersections. In the study of Probability, given two Random variables X and Y, the joint distribution of X and Y is the distribution Conditional probability is the Probability of some event A, given the occurrence of some other event B. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty The most natural choice is the Borel measurable set derived from unions and intersections of intervals. In Mathematics, the Borel algebra is the smallest &sigma-algebra on the Real numbers R containing the intervals, and the Borel However, the larger class of Lebesgue measurable sets proves more useful in practice. In Mathematics, the Lebesgue measure, named after Henri Lebesgue, is the standard way of assigning a Length, Area or Volume to

In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with A probability space, in Probability theory, is the conventional Mathematical model of Randomness. In Mathematics, a σ-algebra (or sigma-algebra) ( Sigma is a Greek letter upper case Σ lower case σ over a set X is a nonempty Under this definition, any subset of the sample space that is not an element of the σ-algebra is not an event, and does not have a probability. With a reasonable specification of the probability space, however, all events of interest will be elements of the σ-algebra.

A note on notation

Even though events are subsets of some sample space Ω, they are often written as propositional formulas involving random variables. In Propositional logic, a propositional formula is a type of syntactic formula which is well formed and has a Truth value. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way For example, if X is a real-valued random variable defined on the sample space Ω, the event

can be written more conveniently as, simply,

This is especially common in formulas for a probability, such as

See also

• Probability
• Complementary event