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.

Every random vector gives rise to a probability measure on R < sup > n </ sup > with the Borel algebra as the underlying sigma-algebra.

This is a probability measure defined on the Borel subsets of R

In this case, the probability P ( Y = y ) = 0, and the Borel – Kolmogorov paradox demonstrates the ambiguity of attempting to define conditional probability along these lines.

Then μ is a probability measure on the σ-algebra of Borel subsets of I.

In probability theory, the Borel – Kolmogorov paradox ( sometimes known as Borel's paradox ) is a paradox relating to conditional probability with respect to an event of probability zero ( also known as a null set ).

Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable.

uniquely determines A and conversely, is uniquely determined by A. E < sub > A </ sub > is a boolean homomorphism from the Borel subsets of R into the lattice Q of self-adjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

In mathematics, a subset of a Polish space is universally measurable if it is measurable with respect to every complete probability measure on that measures all Borel subsets of.

Borel, in particular, was careful to point out that negligibility was relative to a model of probability for a specific physical system.

* De Finetti's game – a procedure for evaluating someone's subjective probability

Fourier analysis has many scientific applications – in physics, partial differential equations, number theory, combinatorics, signal processing, imaging, probability theory, statistics, option pricing, cryptography, numerical analysis, acoustics, oceanography, sonar, optics, diffraction, geometry, protein structure analysis and other areas.

While the Trivers – Willard hypothesis predicts that birth sex is dependent on living conditions ( i. e. more male children are born in " good " living conditions, while more female children are born in poorer living conditions ), the probability of having a child of either gender is still regarded as 50 / 50.

What is needed is a hash function H ( z, n ) – where z is the key being hashed and n is the number of allowed hash values – such that H ( z, n + 1 ) = H ( z, n ) with probability close to n /( n + 1 ).

Mutual information can be expressed as the average Kullback – Leibler divergence ( information gain ) of the posterior probability distribution of X given the value of Y to the prior distribution on X:

The Kullback – Leibler divergence ( or information divergence, information gain, or relative entropy ) is a way of comparing two distributions: a " true " probability distribution p ( X ), and an arbitrary probability distribution q ( X ).

In statistics, the Kolmogorov – Smirnov test ( K – S test ) is a nonparametric test for the equality of continuous, one-dimensional probability distributions that can be used to compare a sample with a reference probability distribution ( one-sample K – S test ), or to compare two samples ( two-sample K – S test ).

The Kolmogorov – Smirnov test may also be used to test whether two underlying one-dimensional probability distributions differ.

While the Kolmogorov – Smirnov test is usually used to test whether a given F ( x ) is the underlying probability distribution of F < sub > n </ sub >( x ), the procedure may be inverted to give confidence limits on F ( x ) itself.

* 2004 – Male Po ' ouli ( Black-faced honeycreeper ) dies of Avian malaria in the Maui Bird Conservation Center in Olinda, Hawaii before it could breed, making the species in all probability extinct.

An alternative account of probability emphasizes the role of prediction – predicting future observations on the basis of past observations, not on unobservable parameters.

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 electron might move to a place and time E where it absorbs the photon ; then move on before emitting another photon at F ; then move on to C where it is detected, while the new photon moves on to D. The probability of this complex process can again be calculated by knowing the probabilities of each of the individual actions: three electron actions, two photon actions and two vertexes – one emission and one absorption.

The basic rule is that if we have the probability amplitude for a given complex process involving more than one electron, then when we include ( as we always must ) the complementary Feynman diagram in which we just exchange two electron events, the resulting amplitude is the reverse – the negative – of the first.

* Glivenko – Cantelli theorem in probability