Probability in the context of Measure theory

In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probabilities that each of the states of nature will occur, (e.g. Rain: 0.70, No Rain: 0.30). Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.

In economics, individuals are assumed to rank lotteries according to a rational system of preferences, although it is now accepted that people make irrational choices systematically. Behavioral economics studies what happens in markets in which some of the agents display human complications and limitations.

Justification (also called epistemic justification) is a property of beliefs that fulfill certain norms about what a person should believe. Epistemologists often identify justification as a component of knowledge distinguishing it from mere true opinion. They study the reasons why someone holds a belief. Epistemologists are concerned with various features of belief, which include the ideas of warrant (a proper justification for holding a belief), knowledge, rationality, and probability, among others.

Debates surrounding epistemic justification often involve the structure of justification, including whether there are foundational justified beliefs or whether mere coherence is sufficient for a system of beliefs to qualify as justified. Another major subject of debate is the sources of justification, which might include perceptual experience (the evidence of the senses), reason, and authoritative testimony, among others.

Pierre de Fermat (/fɜːrˈmɑː/; French: [pjɛʁ də fɛʁma]; 17 August 1601 – 12 January 1665) was a French magistrate, polymath, and above all mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica. He was also a lawyer at the parlement of Toulouse, France, a poet, a skilled Latinist, and a Hellenist.

This list contains selected positive numbers in increasing order, including counts of things, dimensionless quantities and probabilities. Each number is given a name in the short scale, which is used in English-speaking countries, as well as a name in the long scale, which is used in some of the countries that do not have English as their national language.

The future is the time after the past and present. Its arrival is considered inevitable due to the existence of time and the laws of physics. Due to the apparent nature of reality and the unavoidability of the future, everything that currently exists and will exist can be categorized as either permanent, meaning that it will exist forever, or temporary, meaning that it will end. In the Occidental view, which uses a linear conception of time, the future is the portion of the projected timeline that is anticipated to occur. In special relativity, the future is considered absolute future, or the future light cone.

In the philosophy of time, presentism is the belief that only the present exists and the future and the past are unreal. Religions consider the future when they address issues such as karma, life after death, and eschatologies that study what the end of time and the end of the world will be. Religious figures such as prophets and diviners have claimed to see into the future.Future studies, or futurology, is the science, art, and practice of postulating possible futures. Modern practitioners stress the importance of alternative and plural futures, rather than one monolithic future, and the limitations of prediction and probability, versus the creation of possible and preferable futures. Predeterminism is the belief that the past, present, and future have been already decided.

Acatalepsy (from the Greek α̉- 'privative' and καταλαμβάνειν 'to seize'), in philosophy, is incomprehensibleness, or the impossibility of comprehending or conceiving some or all things. The doctrine held by the ancient Skeptic philosophers, that human knowledge never amounts to certainty, but only to probability.

The Pyrrhonians attempted to show, while Academic skeptics of the Platonic Academy asserted an absolute acatalepsia; all human science or knowledge, according to them, went no further than to appearances and verisimilitude. It is the antithesis of the Stoic doctrine of katalepsis or Apprehension. According to the Stoics, katalepsis was true perception, but to the Skeptics, all perceptions were acataleptic, i.e. bear no conformity to the objects perceived, or, if they did bear any conformity, it could never be known.

In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events 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 0.5 for X = tails (assuming that the coin is fair). More commonly, probability distributions are used to compare the relative occurrence of many different random values.

In common usage, randomness is the apparent or actual lack of definite patterns or predictability in information. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual random events are, by definition, unpredictable, but if there is a known probability distribution, the frequency of different outcomes over repeated events (or "trials") is predictable. For example, when throwing two dice, the outcome of any particular roll is unpredictable, but a sum of 7 will tend to occur twice as often as 4. In this view, randomness is not haphazardness; it is a measure of uncertainty of an outcome. Randomness applies to concepts of chance, probability, and information entropy.

The fields of mathematics, probability, and statistics use formal definitions of randomness, typically assuming that there is some 'objective' probability distribution. In statistics, a random variable is an assignment of a numerical value to each possible outcome of an event space. This association facilitates the identification and the calculation of probabilities of the events. Random variables can appear in random sequences. A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness.

In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general.

The intuition behind this concept dates back to Ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Constantin Carathéodory, and Maurice Fréchet, among others. According to Thomas W. Hawkins Jr., "It was primarily through the theory of multiple integrals and, in particular the work of Camille Jordan that the importance of the notion of measurability was first recognized."

A Venn diagram is a widely used diagram style that shows the logical relation between sets, popularized by John Venn (1834–1923) in the 1880s. The diagrams are used to teach elementary set theory, and to illustrate simple set relationships in probability, logic, statistics, linguistics and computer science. A Venn diagram uses simple closed curves on a plane to represent sets. The curves are often circles or ellipses.

Similar ideas had been proposed before Venn such as by Christian Weise in 1712 (Nucleus Logicoe Wiesianoe) and Leonhard Euler in 1768 (Letters to a German Princess). The idea was popularised by Venn in Symbolic Logic, Chapter V "Diagrammatic Representation", published in 1881.

Indeterminism is the idea that events (or certain events, or events of certain types) are not caused, or are not caused deterministically.

It is the opposite of determinism and related to chance. It is highly relevant to the philosophical problem of free will, particularly in the form of metaphysical libertarianism. In science, most specifically quantum theory in physics, indeterminism is the belief that no event is certain and the entire outcome of anything is probabilistic. Heisenberg's uncertainty principle and the "Born rule", proposed by Max Born, are often starting points in support of the indeterministic nature of the universe. Indeterminism is also asserted by Sir Arthur Eddington, and Murray Gell-Mann. Indeterminism has been promoted by the French biologist Jacques Monod's essay "Chance and Necessity".The physicist-chemist Ilya Prigogine argued for indeterminism in complex systems.

Heuristics (from Ancient Greek εὑρίσκω (heurískō) 'to find, discover') is the process by which humans use mental shortcuts to arrive at decisions. Heuristics are simple strategies that humans, animals, organizations, and even machines use to quickly form judgments, make decisions, and find solutions to complex problems. Often this involves focusing on the most relevant aspects of a problem or situation to formulate a solution. While heuristic processes are used to find the answers and solutions that are most likely to work or be correct, they are not always right or the most accurate. Judgments and decisions based on heuristics are simply good enough to satisfy a pressing need in situations of uncertainty, where information is incomplete. In that sense they can differ from answers given by logic and probability.

The economist and cognitive psychologist Herbert A. Simon introduced the concept of heuristics in the 1950s, suggesting there were limitations to rational decision making. In the 1970s, psychologists Amos Tversky and Daniel Kahneman added to the field with their research on cognitive bias. It was their work that introduced specific heuristic models, a field which has only expanded since. While some argue that pure laziness is behind the heuristics process, this could just be a simplified explanation for why people don't act the way we expected them to. Other theories argue that it can be more accurate than decisions based on every known factor and consequence, such as the less-is-more effect.

Probability 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. Any specified subset of the sample space is called an event.

Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion).Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability theory describing such behaviour are the law of large numbers and the central limit theorem.

The propensity theory of probability is a probability interpretation in which the probability is thought of as a physical propensity, disposition, or tendency of a given type of situation to yield an outcome of a certain kind, or to yield a long-run relative frequency of such an outcome.

Propensities are not relative frequencies, but purported causes of the observed stable relative frequencies. Propensities are invoked to explain why repeating a certain kind of experiment will generate a given outcome type at a persistent rate. Stable long-run frequencies are a manifestation of invariant single-case probabilities. Frequentists are unable to take this approach, since relative frequencies do not exist for single tosses of a coin, but only for large ensembles or collectives. These single-case probabilities are known as propensities or chances.

Luck is a phenomenon or belief that humans may associate with experiencing improbable events, especially improbably positive or negative events. Philosophical naturalism, eschewing any supernatural explanations, might suggest that positive or negative events may happen at any time (due to both random and non-random natural and artificial processes), and that even improbable events can happen by random chance.In this view, the epithet "lucky" or "unlucky" is a descriptive label that refers to an event's positivity, negativity, or improbability.

Supernatural interpretations of luck consider it to be an attribute of a personor of an object, or the result of a favorable (or unfavorable) view manifested by a deitytowards the lucky (or unlucky) person.These interpretations often prescribe how luckiness or unluckiness can be obtained, such as by carrying a lucky charm or offering sacrifices or prayers to a deity. Saying someone is "born lucky" may hold different meanings, depending on the interpretation: it could simply mean that they have been born into a good family or circumstance; or that they habitually experience improbably positive events, due to some inherent property, or due to the lifelong favor of a god or goddess in a monotheistic or polytheistic religion.

In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Common quantiles have special names, such as quartiles (four groups), deciles (ten groups), and percentiles (100 groups). The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.

q-quantiles are values that partition a finite set of values into q subsets of (nearly) equal sizes. There are q − 1 partitions of the q-quantiles, one for each integer k satisfying 0 < k < q. In some cases the value of a quantile may not be uniquely determined, as can be the case for the median (2-quantile) of a uniform probability distribution on a set of even size. Quantiles can also be applied to continuous distributions, providing a way to generalize rank statistics to continuous variables (see percentile rank). When the cumulative distribution function of a random variable is known, the q-quantiles are the application of the quantile function (the inverse function of the cumulative distribution function) to the values {1/q, 2/q, …, (q − 1)/q}.