- Event (probability theory)
In

probability theory , an**event**is a set of outcomes (asubset of thesample 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 thepower 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 aprobability space it is possible, and often necessary, 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 card s 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 (anelementary event , of which there are 52, representing the 52 possible cards drawn from the deck), theempty 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 areproper subset s of the sample space that contain multiple elements. So, for example, potential events include:* "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 diagram s. 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 theaxioms of probability , and the definition ofconditional probability can be represented in this fashion.)**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 thenormal distribution the sample space is the set of real numbers or some subset of thereal 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 σ-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.In the general measure-theoretic description of

probability space s, an event may be defined as an element of a selected σ-algebra of subsets of the sample space. 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 formula s involvingrandom variable s. For example, if "X" is a real-valued random variable defined on the sample space Ω, the event:$\{omega\; |\; u\; <\; X(omega)\; leq\; v\},$can be written more conveniently as, simply,:$u\; <\; X\; leq\; v,.$This is especially common in formulas for aprobability , such as:$P(u\; <\; X\; leq\; v)\; =\; F(v)-F(u),.$**ee also***

Probability

*Complementary event

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Probability theory**— is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non deterministic events or measured… … Wikipedia**probability theory**— Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… … Universalium**Martingale (probability theory)**— For the martingale betting strategy , see martingale (betting system). Stopped Brownian motion is an example of a martingale. It can be used to model an even coin toss betting game with the possibility of bankruptcy. In probability theory, a… … Wikipedia**Independence (probability theory)**— In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. For example: The event of getting a 6 the first time a die is rolled… … Wikipedia**Probability**— is the likelihood or chance that something is the case or will happen. Probability theory is used extensively in areas such as statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the… … Wikipedia**Probability space**— This article is about mathematical term. For the novel, see Probability Space (novel). In probability theory, a probability space or a probability triple is a mathematical construct that models a real world process (or experiment ) consisting of… … Wikipedia**Probability interpretations**— The word probability has been used in a variety of ways since it was first coined in relation to games of chance. Does probability measure the real, physical tendency of something to occur, or is it just a measure of how strongly one believes it… … Wikipedia**Probability distribution**— This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics). For other uses, see Distribution (disambiguation). In probability theory, a probability mass, probability density … Wikipedia**probability and statistics**— ▪ mathematics Introduction the branches of mathematics concerned with the laws governing random events, including the collection, analysis, interpretation, and display of numerical data. Probability has its origin in the study of gambling… … Universalium**Theory**— The word theory has many distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion.In science a theory is a testable model of the manner of interaction of a set of natural phenomena,… … Wikipedia