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.
Sample spaceIn probability theory, the sample space (also called sample description space, possibility space, or outcome 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, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols.
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.
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 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 .
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.
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).
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.
Outcome (probability)In probability theory, an outcome is a possible result of an experiment or trial. Each possible outcome of a particular experiment is unique, and different outcomes are mutually exclusive (only one outcome will occur on each trial of the experiment). All of the possible outcomes of an experiment form the elements of a sample space. For the experiment where we flip a coin twice, the four possible outcomes that make up our sample space are (H, T), (T, H), (T, T) and (H, H), where "H" represents a "heads", and "T" represents a "tails".
AxiomAn axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.
Event (probability theory)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. A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. An event consisting of only a single outcome is called an or an ; that is, it is a singleton set. An event is said to if contains the outcome of the experiment (or trial) (that is, if ).
Peano axiomsIn mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th-century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.
Tarski's axiomsTarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory (i.e., that part of Euclidean geometry that is formulable as an elementary theory). Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms. Early in his career Tarski taught geometry and researched set theory.
Many-worlds interpretationThe many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is objectively real, and that there is no wave function collapse. This implies that all possible outcomes of quantum measurements are physically realized in some "world" or universe. In contrast to some other interpretations, such as the Copenhagen interpretation, the evolution of reality as a whole in MWI is rigidly deterministic and local.
Separation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are not fundamental axioms like those of set theory, but rather defining properties which may be specified to distinguish certain types of topological spaces.
Hilbert's axiomsHilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
Ensemble interpretationThe ensemble interpretation of quantum mechanics considers the quantum state description to apply only to an ensemble of similarly prepared systems, rather than supposing that it exhaustively represents an individual physical system. The advocates of the ensemble interpretation of quantum mechanics claim that it is minimalist, making the fewest physical assumptions about the meaning of the standard mathematical formalism. It proposes to take to the fullest extent the statistical interpretation of Max Born, for which he won the Nobel Prize in Physics in 1954.
Kuratowski closure axiomsIn topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formalized by Kazimierz Kuratowski, and the idea was further studied by mathematicians such as Wacław Sierpiński and António Monteiro, among others. A similar set of axioms can be used to define a topological structure using only the dual notion of interior operator.
