Sample areas in the new investigations were selected strictly by application of the principles of probability theory, so as to be representative of the total population of defined areas within calculable limits.

This list could be expanded to include most fields of mathematics, including measure theory, ergodic theory, probability, representation theory, and differential geometry.

Occasionally, " almost all " is used in the sense of " almost everywhere " in measure theory, or in the closely related sense of " almost surely " in probability theory.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

In information theory, one bit is typically defined as the uncertainty of a binary random variable that is 0 or 1 with equal probability, or the information that is gained when the value of such a variable becomes known.

Pascal was an important mathematician, helping create two major new areas of research: he wrote a significant treatise on the subject of projective geometry at the age of sixteen, and later corresponded with Pierre de Fermat on probability theory, strongly influencing the development of modern economics and social science.

In computational complexity theory, BPP, which stands for bounded-error probabilistic polynomial time is the class of decision problems solvable by a probabilistic Turing machine in polynomial time, with an error probability of at most 1 / 3 for all instances.

In computational complexity theory, BQP ( bounded error quantum polynomial time ) is the class of decision problems solvable by a quantum computer in polynomial time, with an error probability of at most 1 / 3 for all instances.

Following the work on expected utility theory of Ramsey and von Neumann, decision-theorists have accounted for rational behavior using a probability distribution for the agent.

Johann Pfanzagl completed the Theory of Games and Economic Behavior by providing an axiomatization of subjective probability and utility, a task left uncompleted by von Neumann and Oskar Morgenstern: their original theory supposed that all the agents had the same probability distribution, as a convenience.

The " Ramsey test " for evaluating probability distributions is implementable in theory, and has kept experimental psychologists occupied for a half century.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

In part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc.

Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis and probability theory.

In probability theory and statistics, the cumulative distribution function ( CDF ), or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x.

This is totally spurious, since no matter who measured first the other will measure the opposite spin despite the fact that ( in theory ) the other has a 50 % ' probability ' ( 50: 50 chance ) of measuring the same spin, unless data about the first spin measurement has somehow passed faster than light ( of course TI gets around the light speed limit by having information travel backwards in time instead ).

In the computer science subfield of algorithmic information theory, a Chaitin constant ( Chaitin omega number ) or halting probability is a real number that informally represents the probability that a randomly constructed program will halt.

* statements assuming a combination of intention and probability (" they are moving ") to distinguish the likely series of events and dependencies (" if they can sell their house they might move to Agrestic or if they can't, to Gardendale ") in which the probability that an attempt to do something complex may fail is explicitly acknowledged and not assumed certain.

One takes probability as ' a degree of rational belief ', or some similar idea ... the second defines probability in terms of frequencies of occurrence of events, or by relative proportions in ' populations ' or ' collectives '; ( p. 101 )

When the probability of different events is not independent, the probability of future events can change based on the outcome of past events ( see statistical permutation ).

The outcome of future events can be affected if external factors are allowed to change the probability of the events ( e. g., changes in the rules of a game affecting a sports team's performance levels ).

When the probability of repeated events are not known, outcomes may not be equally probable.

Amos Tversky and Daniel Kahneman first proposed that the gambler's fallacy is a cognitive bias produced by a psychological heuristic called the representativeness heuristic, which states that people evaluate the probability of a certain event by assessing how similar it is to events they have experienced before, and how similar the events surrounding those two processes are.

Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs, all implicitly assuming events of equal probability.

Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.

The concept has been given an axiomatic mathematical derivation in probability theory, which is used widely in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence / machine learning and philosophy to, for example, draw inferences about the expected frequency of events.

In Kolmogorov's formulation ( see probability space ), sets are interpreted as events and probability itself as a measure on a class of sets.

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 ).

The most important distinction between the frequentist and Bayesian paradigms, is that frequentist makes strong distinctions between probability, statistics, and decision-making, whereas Bayesians unify decision-making, statistics and probability under a single philosophically and mathematically consistent framework, unlike the frequentist paradigm which has been proven to be inconsistent, especially for real-world situations where experiments ( or " random events ") can not be repeated more than once.

The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single occurrences or evolve over time in an apparently random fashion.

If there are g ( E ) dE states with energy E to E + dE, then the Boltzmann distribution predicts a probability distribution for the energy:

* Bertrand's paradox: a paradox in classical probability, solved by E. T.

E. T. Jaynes, from a Bayesian point of view, pointed out probability is a measure of a human's information about the physical world.

The formula provides the GoS ( grade of service ) which is the probability P < sub > b </ sub > that a new call arriving at the circuit group is rejected because all servers ( circuits ) are busy: B ( E, m ) when E Erlang of traffic are offered to m trunks ( communication channels ).

In 1972, James E. Nymann showed that the probability that k independently chosen integers are coprime is 1 / ζ ( k ).

In 1917, Albert Einstein established the theoretical foundations for the laser and the maser in the paper Zur Quantentheorie der Strahlung ( On the Quantum Theory of Radiation ); via a re-derivation of Max Planck ’ s law of radiation, conceptually based upon probability coefficients ( Einstein coefficients ) for the absorption, spontaneous emission, and stimulated emission of electromagnetic radiation ; in 1928, Rudolf W. Ladenburg confirmed the existences of the phenomena of stimulated emission and negative absorption ; in 1939, Valentin A. Fabrikant predicted the use of stimulated emission to amplify “ short ” waves ; in 1947, Willis E. Lamb and R. C. Retherford found apparent stimulated emission in hydrogen spectra and effected the first demonstration of stimulated emission ; in 1950, Alfred Kastler ( Nobel Prize for Physics 1966 ) proposed the method of optical pumping, experimentally confirmed, two years later, by Brossel, Kastler, and Winter.

Then the probability of the measurement outcome lying in an interval B of R is | E < sub > A </ sub >( B ) ψ |< sup > 2 </ sup >.

This degree of support of H by E has been called the logical probability of H given E, or the epistemic probability of H given E, or the inductive probability of H given E.

In probability theory, the probability P of some event E, denoted, is usually defined in such a way that P satisfies the Kolmogorov axioms, named after the famous Russian mathematician Andrey Kolmogorov, which are described below.

From a knowledge of the probabilities of each of these subprocesses – E ( A to C ) and P ( B to D ) – then we would expect to calculate the probability of both happening by multiplying them, using rule b ) above.

The P < small >< sub > k </ sub ></ small > ( kill probability ) of the AIM-7E was less than 10 %; US fighter pilots shot down 55 aircraft using the Sparrow.

One isotope of cadmium, < sup > 113 </ sup > Cd, absorbs neutrons with very high probability if they have an energy below the cadmium cut-off and transmits them otherwise.

The probability that X lies in the interval ( a, b < nowiki >

The probability measure on Cantor space, sometimes called the fair-coin measure, is defined so that for any binary string x the set of sequences that begin with x has measure 2 < sup >-| x |</ sup >.

In this way, Ω < sub > F </ sub > represents the probability that a randomly selected infinite sequence of 0s and 1s begins with a bit string ( of some finite length ) that is in the domain of F. It is for this reason that Ω < sub > F </ sub > is called a halting probability.

Given the first n digits of Ω and a k ≤ n, the algorithm enumerates the domain of F until enough elements of the domain have been found so that the probability they represent is within 2 < sup >-( k + 1 )</ sup > of Ω.

In a sense that can be made precise, the probability that two randomly chosen integers are coprime is 6 / π < sup > 2 </ sup > ( see pi ), which is about 61 %.

Suppose random variable X can take value x < sub > 1 </ sub > with probability p < sub > 1 </ sub >, value x < sub > 2 </ sub > with probability p < sub > 2 </ sub >, and so on, up to value x < sub > k </ sub > with probability p < sub > k </ sub >.