Important and commonly encountered probability distributions include the binomial distribution, the hypergeometric distribution, and the normal distribution.

The metaphor of defective and drawn objects depicts an application of the hypergeometric distribution in which the interchange symmetry between n and m is not of foremost concern.

For this reason, each of these quadrants is governed by an equivalent hypergeometric distribution.

The symmetries of the hypergeometric distribution provide many options in how to conduct the sampling procedure to isolate the degree of freedom governed by the hypergeometric distribution.

The test ( see above ) based on the hypergeometric distribution ( hypergeometric test ) is identical to the corresponding one-tailed version of Fisher's exact test ).

The probability of drawing any sequence of white and black marbles ( the hypergeometric distribution ) depends only on the number of white and black marbles, not on the order in which they appear ; i. e., it is an exchangeable distribution.

* If the probabilities to draw a white or black marble are not equal ( e. g. because their size is different ) then has a Noncentral hypergeometric distribution

* Survey Analysis Tool based on A. Berkopec, HyperQuick algorithm for discrete hypergeometric distribution, Journal of Discrete Algorithms, Elsevier, 2006.

Other examples of distributions that are not exponential families are the F-distribution, Cauchy distribution, hypergeometric distribution and logistic distribution.

* hypergeometric distribution: the balls are not returned to the urn once extracted.

* multivariate hypergeometric distribution: as above, but with balls of more than two colors.

To evaluate the " complete ignorance " case when s = 0 or s = n can be dealt with by first going back to the hypergeometric distribution, denoted by.

As pointed out by Fisher, this leads under a null hypothesis of independence to a hypergeometric distribution of the numbers in the cells of the table.

The resulting probability distribution is the hypergeometric distribution.

The Bessel functions can be expressed in terms of the generalized hypergeometric series as

Many mathematical theorems can be reduced to more straightforward computation, including polynomial identities, trigonometric identities and hypergeometric identities.

The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function ( Kummer's function ):

The probability that one of the next two cards turned is a club can be calculated using hypergeometric with k = 1, n = 2, m = 9 and N = 47.

The probability that both of the next two cards turned are clubs can be calculated using hypergeometric with k = 2, n = 2, m = 9 and N = 47.

The probability that neither of the next two cards turned are clubs can be calculated using hypergeometric with k = 0, n = 2, m = 9 and N = 47.

Reciprocally, the p-value of a two-sided Fisher's exact test can be calculated as the sum of two appropriate hypergeometric tests ( for more information see ).

Like all such equations, it can be converted into the hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.

Many of the other formulas for special values of hypergeometric functions can be derived from this as special or limiting cases.

A wide range of integrals of elementary functions can be expressed using the hypergeometric function, e. g.:

Their importance and role can be understood through the following example: the hypergeometric series < sub > 2 </ sub > F < sub > 1 </ sub > has the Legendre polynomials as a special case, and when considered in the form of spherical harmonics, these polynomials reflect, in a certain sense, the symmetry properties of the two-sphere or, equivalently, the rotations given by the Lie group SO ( 3 ).

In tensor product decompositions of concrete representations of this group Clebsch-Gordan coefficients are met, which can be written as < sub > 3 </ sub > F < sub > 2 </ sub > hypergeometric series.

Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.

The ordinary differential equation is equivalent to the hypergeometric differential equation and f ( z ) can be written in terms of hypergeometric functions.

( For more on umbral variants of the binomial theorem, see binomial type ) The Chu – Vandermonde identity can also be seen to be a special case of Gauss's hypergeometric theorem, which states that

Since the ratio of coefficients is a rational function, the power series can be written as a generalized hypergeometric series.

The formula for can be expressed in terms of the volume of the unit n-ball and the hypergeometric function or the regularized incomplete beta function as

which can be further simplified by the use of hypergeometric series.

It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hypergeometric functions.

Many linear homogeneous recurrence relations may be solved by means of the generalized hypergeometric series.

Nevertheless, purely real expressions of the solutions may be obtained using hypergeometric functions, or more elementarily in terms of trigonometric functions, specifically in terms of the cosine and arccosine functions.

In mathematics, a generalized hypergeometric series is a series in which the ratio of successive coefficients indexed by n is a rational function of n. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation.

The ordinary hypergeometric series should not be confused with the basic hypergeometric series, which, despite its name, is a rather more complicated and recondite series.

However in the theory of special functions ( in particular the hypergeometric function ) the Pochhammer symbol is used to represent the rising factorial.

The term Generalized hypergeometric function is used for the functions if there is risk of confusion.

The term " hypergeometric series " was first used by John Wallis in his 1655 book Arithmetica Infinitorum.

If we determine the number of hands that contain two red cards, by symmetry relations we will necessarily also know the hypergeometric distributions governing the other three quadrants: hand counts for red / black, black / red, and black / black.