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.

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.

* 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 ).

* Bertrand's paradox: a paradox in classical probability, solved by E. T.

The Fermi paradox ( or Fermi's paradox ) is the apparent contradiction between high estimates of the probability of the existence of extraterrestrial civilization and humanity's lack of contact with, or evidence for, such civilizations.

The Fermi paradox is a conflict between an argument of scale and probability and a lack of evidence.

* Bertrand paradox ( probability )

Stated simply, the Novikov consistency principle asserts that if an event exists that would give rise to a paradox, or to any " change " to the past whatsoever, then the probability of that event is zero.

The self-consistency principle guarantees that the sequence of events generating the paradox in the nested conditional has zero probability.

This work described several now famous results, including Condorcet's jury theorem, which states that if each member of a voting group is more likely than not to make a correct decision, the probability that the highest vote of the group is the correct decision increases as the number of members of the group increases, and Condorcet's paradox, which shows that majority preferences become intransitive with three or more options – it is possible for a certain electorate to express a preference for A over B, a preference for B over C, and a preference for C over A, all from the same set of ballots.

* Lottery paradox ( probability )

* Sleeping beauty paradox ( probability )

In probability theory, the birthday problem or birthday paradox concerns the probability that, in a set of n randomly chosen people, some pair of them will have the same birthday.

With random UUIDs, the chance of two having the same value can be calculated using probability theory ( Birthday paradox ).

According to the birthday paradox, in a group of 23 ( or more ) randomly chosen people, the probability is more than 50 % that some pair of them will have the same birthday.

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.

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 ).

In probability and statistics, the Yule – Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon.

The probability mass function of the Yule – Simon ( ρ ) distribution is

The probability mass function of the generalized Yule – Simon ( ρ, α ) distribution is defined as

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

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 ).

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.