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.

Archaeoastronomy uses a variety of methods to uncover evidence of past practices including archaeology, anthropology, astronomy, statistics and probability, and history.

covers statistical study, descriptive statistics ( collection, description, analysis, and summary of data ), probability, and the binomial and normal distributions, test of hypotheses and confidence intervals, linear regression, and correlation.

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

For objectivists, probability objectively measures the plausibility of propositions, i. e. the probability of a proposition corresponds to a reasonable belief everyone ( even a " robot ") sharing the same knowledge should share in accordance with the rules of Bayesian statistics, which can be justified by requirements of rationality and consistency.

After the 1920s, " inverse probability " was largely supplanted by a collection of methods that came to be called frequentist statistics.

* Conjugate prior, in Bayesian statistics, a family of probability distributions that contains a prior and the posterior distributions for a particular likelihood function ( particularly for one-parameter exponential families )

It has applications that include probability, statistics, computer vision, image and signal processing, electrical engineering, and differential equations.

This generally means that descriptive statistics, unlike inferential statistics, are not developed on the basis of probability theory.

As with other branches of statistics, experimental design is pursued using both frequentist and Bayesian approaches: In evaluating statistical procedures like experimental designs, frequentist statistics studies the sampling distribution while Bayesian statistics updates a probability distribution on the parameter space.

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.

* In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions.

The gamma function is a component in various probability-distribution functions, and as such it is applicable in the fields of probability and statistics, as well as combinatorics.

* fundamental applications of probability and statistics

Information theory is based on probability theory and statistics.

The most complicated aspect of the insurance business is the actuarial science of ratemaking ( price-setting ) of policies, which uses statistics and probability to approximate the rate of future claims based on a given risk.

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 ).

We repeat, that the test of a violation of 7 is whether, at the time of suit, there is a reasonable probability that the acquisition is likely to result in the condemned restraints.

In the case of new evidence, there must be a high probability that its presence or absence would have made a material difference in the trial.

If a defendant has been convicted and can prove that his lawyer did not adequately handle his case and that there is a reasonable probability that the result of the trial would have been different had the lawyer given competent representation, he is entitled to a new trial.

* Pattern: A character who has walked the pattern can walk in shadow to any possible universe, and while there can manipulate probability.

In other words, there is an algorithm for a quantum computer ( a quantum algorithm ) that solves the decision problem with high probability and is guaranteed to run in polynomial time.

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

Broadly speaking, there are two views on Bayesian probability that interpret the probability concept in different ways.

Broadly speaking, there are two views on Bayesian probability that interpret the ' probability ' concept in different ways.

In fact, there are non-Bayesian updating rules that also avoid Dutch books ( as discussed in the literature on " probability kinematics " following the publication of Richard C. Jeffrey's rule, which is itself regarded as Bayesian ).

But once it has hit, there is no longer any probability whatsoever that it will hit somewhere else.

Each halting probability is a normal and transcendental real number which is not computable, which means that there is no algorithm that enumerates its digits.

More exciting for planetary astronomers was that the best orbital solutions suggested that the comet would pass within of the center of Jupiter, a distance smaller than the planet's radius, meaning that there was an extremely high probability that SL9 would collide with Jupiter in July 1994.

In fact, if there are atomic centers in an area of the plane, this probability is, which is simply the ratio of the aggregate area of circles of radius drawn around the points to the whole area.

Also note that if there is a cancellation of waves at some point, that does not mean that a photon disappears ; it only means that the probability of a photon's appearing at that point will decrease, and the probability that it will appear somewhere else increases.

If it is " observed " ( measured with a photon ) not at a particular slit but rather at the screen, then there is no " which path " information as part of the interaction, so the electron's " observed " position on the screen is determined strictly by its probability function.

* " IF the identity of the germ is not known with certainty AND the germ is gram-positive AND the morphology of the organism is " rod " AND the germ is aerobic THEN there is a strong probability ( 0. 8 ) that the germ is of type enterobacteriacae "

In July of that year, Fermi submitted his doctoral thesis Un teorema di calcolo delle probabilità ed alcune sue applicazioni ( A theorem on probability and some of its applications ) to the Scuola Normale Superiore and received his Laurea from there at the unusually young age of 21.

For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked ( i. e., rejected ) due to all circuits being in use ( a GoS of 0. 01 ), which becomes the target probability of call blocking, P < sub > b </ sub >, when using the Erlang B formula.

In this case, if Bob subsequently measures spin along the z-axis, there is 100 % probability that he will obtain − z.

Therefore, in the one measurement he is allowed to make, there is a 50 % probability of getting "+" and 50 % of getting "−", regardless of whether or not his axis is aligned with Alice's.

In other words, the previous losses in no way contribute to the odds of the remaining attempts, but there are fewer remaining attempts to gain a win, which results in a lower probability of obtaining it.

For any information rate R < C and coding error ε > 0, for large enough N, there exists a code of length N and rate ≥ R and a decoding algorithm, such that the maximal probability of block error is ≤ ε ; that is, it is always possible to transmit with arbitrarily small block error.