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.

If μ is less than 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction with probability 1 by Markov's inequality.

In mathematics, the Cauchy – Schwarz inequality ( also known as the Bunyakovsky inequality, the Schwarz inequality, or the Cauchy – Bunyakovsky – Schwarz inequality, or Cauchy – Bunyakovsky inequality ), is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas.

For a 90 % probability, covering the range from the 5 % to the 95 % range on the probability curve, the upper and lower limits can be found using the inequality:

The Chebyshev inequality states that if is a random variable with standard deviation σ, then the probability that the outcome of is no less than away from its mean is no more than:

* Chebyshev's inequality in probability and statistics

In probability theory, Chebyshev ’ s inequality ( also spelled as Tchebysheff ’ s inequality ) guarantees that in any probability distribution ," nearly all " values are close to the mean — the precise statement being that no more than 1 / k < sup > 2 </ sup > of the distribution ’ s values can be more than k standard deviations away from the mean.

When the right-hand side is greater than one, so the inequality becomes vacuous, as the probability of any event cannot be greater than one.

We can then infer that the probability that it has between 600 and 1400 words ( i. e. within k = 2 SDs of the mean ) must be more than 75 %, because there is less than chance to be outside that range, by Chebyshev ’ s inequality.

This example should be treated with caution as the inequality is only stated for probability distributions rather than for finite sample sizes.

Chebyshev's inequality uses the variance to bound the probability that a random variable deviates far from the mean.

The inequality can be stated quite generally using either the language of measure theory or ( equivalently ) probability.

In probability theory, Boole's inequality, also known as the union bound, says that for any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.

In measure-theoretic terms, Boole's inequality follows from the fact that a measure ( and certainly any probability measure ) is σ-sub-additive.

Boole's inequality may be generalised to find upper and lower bounds on the probability of finite unions of events.

Alternatively, for a single system at a well-defined temperature, it gives the probability that the system is in the specified state.

The term or, which gives the ( unnormalised ) relative probability of a state, is called the Boltzmann factor and appears often in the study of physics and chemistry.

Copying the numbers that won the previous lottery draw gives an equal probability, although a rational gambler might attempt to predict other players ' choices and then deliberately avoid these numbers.

For example, consider a model which gives the probability density function of observable random variable X as a function of a parameter θ.

If the random variable is real-valued ( or more generally, if a total order is defined for its possible values ), the cumulative distribution function gives the probability that the random variable is no larger than a given value ; in the real-valued case it is the integral of the density.

Thus, measuring a quantum state described by complex coefficients ( a, b ,..., h ) gives the classical probability distribution and we say that the quantum state " collapses " to a classical state as a result of making the measurement.

Fermat's principle is the main principle of quantum electrodynamics where it states that any particle ( e. g. a photon or an electron ) propagates over all available ( unobstructed ) paths and the interference ( sum, or superposition ) of its wavefunction over all those paths ( at the point of observer or detector ) gives the correct probability of detection of this particle ( at this point ).

Good gives an example of background knowledge with respect to which the observation of a black raven decreases the probability that all ravens are black:

Barlow and Proschan define availability of a repairable system as " the probability that the system is operating at a specified time t ." While Blanchard gives a qualitative definition of availability as " a measure of the degree of a system which is in the operable and committable state at the start of mission when the mission is called for at an unknown random point in time.

The probability of congestion gives the Grade of Service experienced.

As Kittel and Kroemer put it, " The probability of Hamlet is therefore zero in any operational sense of an event ...", and the statement that the monkeys must eventually succeed " gives a misleading conclusion about very, very large numbers.

There may be cases where the match probability in relation to all the samples tested is so great that the judge would consider its probative value to be minimal and decide to exclude the evidence in the exercise of his discretion, but this gives rise to no new question of principle and can be left for decision on a case by case basis.

For k + r Bernoulli trials with success probability p, the negative binomial gives the probability of k successes and r failures, with a failure on the last trial.

", this gives the probability.

" this gives the probability.

For sufficiently nice prior probabilities, the Bernstein-von Mises theorem gives that in the limit of infinite trials and the posterior converges to a Gaussian distribution independent of the initial prior under some conditions firstly outlined and rigorously proven by Joseph Leo Doob in 1948, namely if the random variable in consideration has a finite probability space.

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

Though multiple accidental ( SIDS ) deaths are rare, so are multiple murders ; with only the facts of the deaths as evidence, it is the ratio of these ( prior ) improbabilities that gives the correct " posterior probability " of murder.

By repeated random selection of a possible witness, the large probability that a random string is a witness gives an expected polynomial time algorithm for accepting or rejecting an input.

In the later editions of his book, de Moivre gives the first statement of the formula for the normal distribution curve, the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the probable error calculation.

Like many quantum algorithms, Grover's algorithm is probabilistic in the sense that it gives the correct answer with high probability.

If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is.

The quantities that describe how they change in time are the transition probabilities, which gives the probability that, starting at x, the particle ends up at y after time t. The total probability of ending up at y is given by the sum over all the possibilities