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 the response variable is expected to follow a parametric family of probability distributions, then the statistician may specify ( in the protocol for the experiment or observational study ) that the responses be transformed to stabilize the variance.

Various results in probability theory about expected values, such as the strong law of large numbers, will not work in such cases.

From a rigorous theoretical standpoint, the expected value is the integral of the random variable with respect to its probability measure.

Laplace wrote of the ways men calculated their probability of having sons: " I have seen men, ardently desirous of having a son, who could learn only with anxiety of the births of boys in the month when they expected to become fathers.

: The expected value ( in the sense of probability theory ) of the observable A for the system in state represented by the unit vector H is

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.

Pot odds are often compared to the probability of winning a hand with a future card in order to estimate the call's expected value.

More Probability density function | probability density will be found the closer one gets to the expected ( mean ) value in a normal distribution.

Cumulative probability of a normal distribution with expected value 0 and standard deviation 1

In statistics and probability theory, standard deviation ( represented by the symbol sigma, σ ) shows how much variation or " dispersion " exists from the average ( mean, or expected value ).

It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean ( expected value ).

By the law of large numbers, the sample averages converge in probability and almost surely to the expected value µ as n tends to infinity.

A reproducibility limit is the value below which the difference between two test results obtained under reproducibility conditions may be expected to occur with a probability of approximately 0. 95 ( 95 %).

Often, although the bounds do exist, they can be safely ignored because the differences between the real-world and theory is not statistically significant, as the probability that such boundary situations might occur is remote compared to the expected normal situation.

For other alignments, we expect some results to be 1 and some to be-1 with a probability that depends on the expected alignment.

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 laboratory, uncertainty is eliminated and calculating the expected returns should be a simple mathematical exercise, because participants are endowed with assets that are defined to have a finite lifespan and a known probability distribution of dividends.

For any collection of fixed size, the expected running time of the algorithm is finite for much the same reason that the infinite monkey theorem holds: there is some probability of getting the right permutation, so given an unbounded number of tries it will almost surely eventually be chosen.

The goal then is to minimize the expected loss, with the expectation taken over the probability distribution of.

: The probability of any event is the ratio between the value at which an expectation depending on the happening of the event ought to be computed, and the value of the thing expected upon its happening

In modern utility theory, expected utility can ( with qualifications, because buying risk for small amounts or buying security for big amounts also happen ) be taken as the probability of an event times the payoff received in case of that event.

For a Bernoulli random variable, the expected value is the theoretical probability of success, and the average of n such variables ( assuming they are independent and identically distributed ( i. i. d.

The weak law of large numbers states that the sample average converges in probability towards the expected value < sup >

The computer method calculates the probability ( p-value ) of a value of

A subsequent review of these tests by the Federal Insecticide, Fungicide, and Rodenticide Act Scientific Advisory Panel points out that while " the negative results decrease the probability that the Cry9C protein is the cause of allergic symptoms in the individuals examined ... in the absence of a positive control and questions regarding the sensitivity and specificity of the assay, it is not possible to assign a negative predictive value to this "

In Bayesian statistics, a probability can be assigned to a hypothesis that can differ from 0 or 1 if the truth value is uncertain.

where the right-hand side represents the probability that the random variable X takes on a value less than or

* Reliability ( the probability of spontaneous bit value change under various conditions ).

* Reliability ( the probability of spontaneous bit value change under various conditions ).

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 >.

That is, every hash value in the output range should be generated with roughly the same probability.

), and each input may independently occur with uniform probability, then a hash function need only map roughly the same number of inputs to each hash value.

If these bits are known ahead of transmission ( to be a certain value with absolute probability ), logic dictates that no information has been transmitted.

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: