Conditional probabilityIn probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. 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(AB) or occasionally P_B(A).
ProbabilityProbability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.
Bayesian probabilityBayesian probability (ˈbeɪziən or ˈbeɪʒən ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown.
Probability theoryProbability theory or probability calculus 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.
Probability distributionIn probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.
Probability axiomsThe Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem. The assumptions as to setting up the axioms can be summarised as follows: Let be a measure space with being the probability of some event , and .
MarriageMarriage, also called matrimony or wedlock, is a culturally and often legally recognized union between people called spouses. It establishes rights and obligations between them, as well as between them and their children (if any), and between them and their in-laws. It is nearly a cultural universal, but the definition of marriage varies between cultures and religions, and over time. Typically, it is an institution in which interpersonal relationships, usually sexual, are acknowledged or sanctioned.
Probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , which assigns each event in the event space a probability, which is a number between 0 and 1.
Probability interpretationsThe word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories of probability interpretations which can be called "physical" and "evidential" probabilities.
Frequentist probabilityFrequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation.
Civil marriageA civil marriage is a marriage performed, recorded, and recognized by a government official. Such a marriage may be performed by a religious body and recognized by the state, or it may be entirely secular. Every country maintaining a population registry of its residents keeps track of marital status, and all UN Member countries except Iran, Somalia, South Sudan, Sudan, and Tonga have signed or ratified either the United Nations Convention on Consent to Marriage, Minimum Age for Marriage, and Registration of Marriages (1962) or the United Nations Convention on the Elimination of All Forms of Discrimination against Women (1979) which carry a responsibility to register marriages.
Open marriageOpen marriage is a form of non-monogamy in which the partners of a dyadic marriage agree that each may engage in extramarital sexual relationships, without this being regarded by them as infidelity, and consider or establish an open relationship despite the implied monogamy of marriage. There are variant forms of open marriage such as swinging and polyamory, each with the partners having varying levels of input into their spouse's activities.
Child marriageChild marriage is a marriage, similar union, formal or informal, between a child under a certain age – typically 18 years – and an adult or another child. The vast majority of child marriages are between a female child and a male adult, and are rooted in gender inequality. Although the age of majority (legal adulthood) and marriageable age are usually designated at age 18, both vary across countries, and therefore the marriageable age may be older or younger in a given country.
Common-law marriageCommon-law marriage, also known as non-ceremonial marriage, sui iuris marriage, informal marriage, de facto marriage, or marriage by habit and repute. In the jurisdictions where marriage can still be contracted this way, a common law marriage is a legal marriage despite non-compliance with the requirements for a statutory marriage. The original concept of a "common-law marriage" is one considered valid by both partners, but not formally recorded with a state or religious registry, nor celebrated in a formal civil or religious service.
Christian views on marriageFrom the earliest days of the Christian faith, Christians have viewed marriage as a divinely blessed, lifelong, monogamous union between a man and a woman. However, while many Christians might agree with the traditional definition, the terminology and theological views of marriage have varied through time in different countries, and among Christian denominations. Many Protestants consider marriage to be a sacred institution or "holy ordinance" of God. Catholic and Eastern Orthodox Christians consider marriage as a holy sacrament or sacred mystery.
Bellman equationA Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices. This breaks a dynamic optimization problem into a sequence of simpler subproblems, as Bellman's “principle of optimality" prescribes.
Dynamic programmingDynamic programming is both a mathematical optimization method and an algorithmic paradigm. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively.
Stochastic processIn probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
Decision-makingIn psychology, decision-making (also spelled decision making and decisionmaking) is regarded as the cognitive process resulting in the selection of a belief or a course of action among several possible alternative options. It could be either rational or irrational. The decision-making process is a reasoning process based on assumptions of values, preferences and beliefs of the decision-maker. Every decision-making process produces a final choice, which may or may not prompt action.
Optimal controlOptimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.
