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.

Section 7 is designed to arrest in its incipiency not only the substantial lessening of competition from the acquisition by one corporation of the whole or any part of the stock of a competing corporation, but also to arrest in their incipiency restraints or monopolies in a relevant market which, as a reasonable probability, appear at the time of suit likely to result from the acquisition by one corporation of all or any part of the stock of any other corporation.

The binomial probability distribution may describe the variation that occurs from one set of trials of such a binomial experiment to another.

Since in practice it is not worth contrasting a zero probability with one that is nearly indistinguishable from zero, he prefers to categorize himself as a " de facto atheist ".

Bayesian probability is one of the different interpretations of the concept of probability and belongs to the category of evidential probabilities.

* For the frequentist a hypothesis is a proposition ( which must be either true or false ), so that the frequentist probability of a hypothesis is either one or zero.

Because of statistical probability and restrictions on the relative diameters of the individual tubes, one of the shells, and thus the whole MWNT, is usually a zero-gap metal.

However, consider the short but never halting algorithm which systematically lists and runs all possible programs ; whenever one of them halts its probability gets added to the output ( initialized by zero ).

An elementary reaction involves a few molecules, usually one or two, because of the low probability for several molecules to meet at a certain time.

The center of an atomic projectile striking this plane has geometrically a definite probability of passing within a certain distance of one of these points.

The probability " wave " can be said to " pass through space " because the probability values that one can compute from its mathematical representation are dependent on time.

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.

If multiple fermions have the same spatial probability distribution, then at least one property of each fermion, such as its spin, must be different.

More precisely, he showed that a random graph on vertices, formed by choosing independently whether to include each edge with probability has, with probability tending to 1 as goes to infinity, at most cycles of length or less, but has no independent set of size Therefore, removing one vertex from each short cycle leaves a smaller graph with girth greater than in which each color class of a coloring must be small and which therefore requires at least colors in any coloring.

With a fair coin, the outcomes in different tosses are statistically independent and the probability of getting heads on a single toss is exactly ( one in two ).

It follows that the probability of getting two heads in two tosses is ( one in four ) and the probability of getting three heads in three tosses is ( one in eight ).

Since the probability of a run of five successive heads is only ( one in thirty-two ), a believer in the gambler's fallacy might believe that this next flip is less likely to be heads than to be tails.

We can see from the above that, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2, 097, 152.

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.

The original probability density may be expressed in terms of the characteristic function, essentially by using the inverse Fourier transform:

However, for well-understood faults the probability that a segment may rupture during the next few decades can be estimated.

* 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.

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.

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

), 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.

Land acquisition and habitat restoration efforts have been initiated in certain areas where there is a relatively high probability that the species may have survived to protect any possible surviving individuals.

Other statistical methods may be used in assessing the probability of future losses.

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.

This may be an extreme form of biased sampling, because certain members of the population are totally excluded from the sample ( that is, they have zero probability of being selected ).

For example, the result of a significance test depends on the probability of a result as extreme or more extreme than the observation, and that probability may depend on the design of the experiment.

Failure probability may influence a manufacture's decisions on a product's warranty.

A statistical copolymer in which the probability of finding a particular type of monomer residue at a particular point in the chain is independent of the types of surrounding monomer residue may be referred to as a truly random copolymer ( 3 ).

Though probability initially may have had lowly motivations, its modern influence and use is widespread ranging from Evidence based medicine, through Six sigma, all the way to the Probabilistically checkable proof and the String theory landscape.

This may be obtained from consideration of whether the required prior probability is greater or lesser than a reference probability associated with an urn model or a thought experiment.

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.

More complex experiments, such as those involving stochastic processes defined in continuous-time, may demand the use of more general probability measures.

A 2008 study analyzing the sexual fantasies of 200 heterosexual men by using the Wilson Sex Fantasy Questionnaire exam, determined that males with a pronounced degree of fetish interest had a greater number of older brothers, a high 2D: 4D digit ratio ( which would indicate excessive prenatal estrogen exposure ), and an elevated probability of being left-handed, suggesting that disturbed hemispheric brain lateralization may play a role in deviant attractions.