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Probability theory |
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In probability theory, an **event** is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.^{ [1] } A single outcome may be an element of many different events,^{ [2] } and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes.^{ [3] } An event defines a complementary event, namely the complementary set (the event *not* occurring), and together these define a Bernoulli trial: did the event occur or not?

**Probability theory** is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

In mathematics, a **set** is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

In probability theory, an **outcome** is a possible result of an experiment. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive. All of the possible outcomes of an experiment form the elements of a sample space.

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 uncountably infinite. So, when defining a probability space it is possible, and often necessary, to exclude certain subsets of the sample space from being events (see * Events in probability spaces *, below).

In probability theory, the **sample space** of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels *S*, Ω, or *U*.

In mathematics, the **power set** of any set *S* is the set of all subsets of *S*, including the empty set and S itself, variously denoted as P(S), 𝒫(*S*), ℘(*S*), *P*(*S*), ℙ(*S*), or, identifying the powerset of *S* with the set of all functions from *S* to a given set of two elements, 2^{S}. In axiomatic set theory, the existence of the power set of any set is postulated by the axiom of power set.

In probability theory, a **probability space** or a **probability triple** is a mathematical construct that models a real-world process consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.

If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome. An event, however, is any subset of the sample space, including any singleton set (an elementary event), the empty set (an impossible event, with probability zero) and the sample space itself (a certain event, with probability one). Other events are proper subsets of the sample space that contain multiple elements. So, for example, potential events include:

A **playing card** is a piece of specially prepared pasteboard, heavy paper, thin cardboard, plastic-coated paper, cotton-paper blend, or thin plastic that is marked with distinguishing motifs. Often the front (face) and back of each card has a finish to make handling easier. They are most commonly used for playing card games, and are also used to perform magic tricks, for cardistry, in card throwing, and for building card houses. Some types of cards such as tarot cards are also used for divination. Playing cards are typically palm-sized for convenient handling, and usually are sold together in a set as a **deck of cards** or **pack of cards**.

In probability theory, an **elementary event** is an event which contains only a single outcome in the sample space. Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponds to precisely one outcome.

In mathematics, the **empty set** is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

- "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. In the situation where each outcome in the sample space Ω is equally likely, the probability of an event *A* is the following formula:

A **Venn diagram** is a diagram that shows *all* possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set. The points inside a curve labelled *S* represent elements of the set *S*, while points outside the boundary represent elements not in the set *S*. This lends to easily read visualizations; for example, the set of all elements that are members of both sets *S* and *T*, *S* ∩ *T*, is represented visually by the area of overlap of the regions *S* and *T*. In Venn diagrams the curves are overlapped in every possible way, showing all possible relations between the sets. They are thus a special case of Euler diagrams, which do not necessarily show all relations. Venn diagrams were conceived around 1880 by John Venn. They are used to teach elementary set theory, as well as illustrate simple set relationships in probability, logic, statistics, linguistics, and computer science.

This rule can readily be applied to each of the example events above.

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 that 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 complementation and countable unions of its members. 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 probability theory, the **normal****distribution** is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be **normally distributed** and is called a **normal deviate**.

In mathematics, a **pathological** phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is **well-behaved**.

In probability theory, **conditional probability** is a measure of the probability of an event occurring given that another event has occurred. If the event of interest is *A* and the event *B* is known or assumed to have occurred, "the conditional probability of *A* given *B*", or "the probability of *A* under the condition *B*", is usually written as P(*A* | *B*), or sometimes *P*_{B}(*A*) or P(*A* / *B*). For example, the probability that any given person has a cough on any given day may be only 5%. But if we know or assume that the person has a cold, then they are much more likely to be coughing. The conditional probability of coughing by the unwell might be 75%, then: P(Cough) = 5%; P(Cough | Sick) = 75%

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. 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* are elements of the σ-algebra.

In mathematical analysis and in probability theory, a **σ-algebra** on a set *X* is a collection Σ of subsets of *X* that includes *X* itself, is closed under complement, and is closed under countable unions.

Even though events are subsets of some sample space Ω, they are often written as predicates or indicators involving random variables. 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

The set *u* < *X* ≤ *v* is an example of an inverse image under the mapping *X* because if and only if .

- ↑ Leon-Garcia, Alberto (2008).
*Probability, statistics and random processes for electrical engineering*. Upper Saddle River, NJ: Pearson. - ↑ Pfeiffer, Paul E. (1978).
*Concepts of probability theory*. Dover Publications. p. 18. ISBN 978-0-486-63677-1. - ↑ Foerster, Paul A. (2006).
*Algebra and trigonometry: Functions and applications, Teacher's edition*(Classics ed.). Upper Saddle River, NJ: Prentice Hall. p. 634. ISBN 0-13-165711-9.

Wikimedia Commons has media related to . Event (probability theory) |

- Hazewinkel, Michiel, ed. (2001) [1994], "Random event",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Formal definition in the Mizar system.

In probability theory and statistics, a **probability distribution** is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In more technical terms, the probability distribution is a description of a random phenomenon in terms of the probabilities of events. For instance, if the random variable X is used to denote the outcome of a coin toss, then the probability distribution of X would take the value 0.5 for *X* = heads, and 0.5 for *X* = tails. Examples of random phenomena can include the results of an experiment or survey.

In probability and statistics, a **random variable**, **random quantity**, **aleatory variable**, or **stochastic variable** is described informally as a variable whose values depend on outcomes of a random phenomenon. The formal mathematical treatment of random variables is a topic in probability theory. In that context, a random variable is understood as a measurable function defined on a probability space whose outcomes are typically real numbers.

In probability theory, two events are **independent**, **statistically independent**, or **stochastically independent** if the occurrence of one does not affect the probability of occurrence of the other. Similarly, two random variables are independent if the realization of one does not affect the probability distribution of the other.

In mathematics, a **Borel set** is any set in a topological space that can be formed from open sets through the operations of countable union, countable intersection, and relative complement. Borel sets are named after Émile Borel.

In probability theory, the **conditional expectation**, **conditional expected value**, or **conditional mean** of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.

In mathematics, a **filtration** is an indexed set of subobjects of a given algebraic structure , with the index running over some index set that is a totally ordered set, subject to the condition that

In mathematics, a **π-system** on a set Ω is a collection *P* of certain subsets of Ω, such that

In probability theory, **random element** is a generalization of the concept of random variable to more complicated spaces than the simple real line. The concept was introduced by Maurice Fréchet (1948) who commented that the “development of probability theory and expansion of area of its applications have led to necessity to pass from schemes where (random) outcomes of experiments can be described by number or a finite set of numbers, to schemes where outcomes of experiments represent, for example, vectors, functions, processes, fields, series, transformations, and also sets or collections of sets.”

In the study of stochastic processes, an **adapted process** is one that cannot "see into the future". An informal interpretation is that *X* is adapted if and only if, for every realisation and every *n*, *X _{n}* is known at time

In the study of stochastic processes in mathematics, a **hitting time** is the first time at which a given process "hits" a given subset of the state space. **Exit times** and **return times** are also examples of hitting times.

In probability and statistics, a **realization**, **observation**, or **observed value**, of a random variable is the value that is actually observed. The random variable itself is the process dictating how the observation comes about. Statistical quantities computed from realizations without deploying a statistical model are often called "empirical", as in empirical distribution function or empirical probability.

In probability theory, a **standard probability space**, also called **Lebesgue–Rokhlin probability space** or just **Lebesgue space** is a probability space satisfying certain assumptions introduced by Vladimir Rokhlin in 1940. Informally, it is a probability space consisting of an interval and/or a finite or countable number of atoms.

In mathematics—specifically, in functional analysis—a **weakly measurable function** taking values in a Banach space is a function whose composition with any element of the dual space is a measurable function in the usual (strong) sense. For separable spaces, the notions of weak and strong measurability agree.

In mathematics, a **real-valued function** is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

In probability theory, the **Doob–Dynkin lemma**, named after Joseph L. Doob and Eugene Dynkin, characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other.

In the theory of stochastic processes, a subdiscipline of probability theory, **filtrations** are used to model the information that is available at a given point and therefore play an important role in the formalization of random processes.

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