In probability theory and statistics, the hypergeometric distribution is a
discrete probability distribution that describes the probability of successes
in draws, without replacement, from a finite population of size that contains
exactly successes, wherein each draw is either a success or a failure. In
contrast, the binomial distribution describes the probability of successes
in draws with replacement. In statistics, the hypergeometric test
uses the hypergeometric distribution to calculate the statistical significance
of having drawn a specific successes from the aforementioned population. The
test is often used to identify which sub-populations are over- or
under-represented in a sample. This test has a wide range of applications. For
example, a marketing group could use the test to understand their customer base
by testing a set of known customers for over-representation of various
demographic subgroups. Definition
The following conditions characterize the hypergeometric distribution:
The result of each draw can be classified into one of two mutually
exclusive categories. The probability of a success changes on
each draw, as each draw decreases the population.
A random variable follows the hypergeometric distribution if its
probability mass function is given by where
is the population size, is the number of success states in the
population, is the number of draws,
is the number of observed successes, is a binomial coefficient.
The pmf is positive when . The pmf satisfies the recurrence
relation with
Combinatorial identities As one would expect, the probabilities
sum up to 1 : This is essentially Vandermonde's
identity from combinatorics. Also note the following identity holds:
This follows from the symmetry of the problem, but it can also be shown by
expressing the binomial coefficients in terms of factorials and rearranging the
latter. Application and example
The classical application of the hypergeometric distribution is sampling
without replacement. Think of an urn with two types of marbles, red ones and
green ones. Define drawing a green marble as a success and drawing a red
marble as a failure. If the variable N describes the number of all marbles in
the urn and K describes the number of green marbles, then N − K corresponds to
the number of red marbles. In this example, X is the random variable whose
outcome is k, the number of green marbles actually drawn in the
experiment. This situation is illustrated by the following contingency
table: Now, assume that there are 5 green and
45 red marbles in the urn. Standing next to the urn, you close your eyes and draw
10 marbles without replacement. What is the probability that exactly 4 of the 10
are green? Note that although we are looking at success/failure, the data are
not accurately modeled by the binomial distribution, because the probability of
success on each trial is not the same, as the size of the remaining population
changes as we remove each marble. This problem is summarized by the
following contingency table: The probability of drawing exactly k
green marbles can be calculated by the formula
Hence, in this example calculate Intuitively we would expect it to be
even more unlikely for all 5 marbles to be green.
As expected, the probability of drawing 5 green marbles is roughly 35 times less
likely than that of drawing 4. = Application to Texas Hold'em Poker =
In Hold'em Poker players make the best hand they can combining the two cards in
their hand with the 5 cards eventually turned up on the table. The deck has 52
and there are 13 of each suit. For this example assume a player has 2 clubs in
the hand and there are 3 cards showing on the table, 2 of which are also clubs.
The player would like to know the probability of one of the next 2 cards
to be shown being a club to complete his flush.
There are 4 clubs showing so there are 9 still unseen. There are 5 cards showing
so there are still unseen. The probability that one of the next two
cards turned is a club can be calculated using hypergeometric with and .
The probability that both of the next two cards turned are clubs can be
calculated using hypergeometric with and .
The probability that neither of the next two cards turned are clubs can be
calculated using hypergeometric with and .
Symmetries Swapping the roles of green and red
marbles: Swapping the roles of drawn and not
drawn marbles: Swapping the roles of green and drawn
marbles: Hypergeometric test
The hypergeometric test uses the hypergeometric distribution to measure
the statistical significance of having drawn a sample consisting of a specific
number of successes from a population of size containing successes. In a
test for over-representation of successes in the sample, the
hypergeometric p-value is calculated as the probability of randomly drawing or
more successes from the population in total draws. In a test for
under-representation, the p-value is the probability of randomly drawing or
fewer successes. = Relationship to Fisher's exact test =
The test based on the hypergeometric distribution is identical to the
corresponding one-tailed version of Fisher's exact test ). Reciprocally, the
p-value of a two-sided Fisher's exact test can be calculated as the sum of two
appropriate hypergeometric tests. Order of draws
The probability of drawing any sequence of white and black marbles 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. As a result, the
probability of drawing a white marble in the draw is
Related distributions Let X ~ Hypergeometric(, , ) and .
If then has a Bernoulli distribution with parameter .
Let have a binomial distribution with parameters and ; this models the number
of successes in the analogous sampling problem with replacement. If and are
large compared to , and is not close to 0 or 1, then and have similar
distributions, i.e., . If is large, and are large compared
to , and is not close to 0 or 1, then where is the standard normal
distribution function If the probabilities to draw a white or
black marble are not equal then has a noncentral hypergeometric distribution
The beta-binomial distribution is a conjugate prior for the hypergeometric
distribution. The following table describes four
distributions related to the number of successes in a sequence of draws:
Multivariate hypergeometric distribution The model of an urn with black and white
marbles can be extended to the case where there are more than two colors of
marbles. If there are Ki marbles of color i in the urn and you take n
marbles at random without replacement, then the number of marbles of each color
in the sample has the multivariate hypergeometric distribution. This has
the same relationship to the multinomial distribution that the hypergeometric
distribution has to the binomial distribution—the multinomial
distribution is the "with-replacement" distribution and the multivariate
hypergeometric is the "without-replacement" distribution.
The properties of this distribution are given in the adjacent table, where c is
the number of different colors and is the total number of marbles.
= Example = Suppose there are 5 black, 10 white, and
15 red marbles in an urn. You reach in and randomly select six marbles without
replacement. What is the probability that you pick exactly two of each color?
Note: When picking the six marbles without replacement, the expected number
of black marbles is 6×(5/30) = 1, the expected number of white marbles is
6×(10/30) = 2, and the expected number of red marbles is 6×(15/30) = 3. This
comes from the expected value of a Binomial distribution, E(X) = np.
See also Multinomial distribution
Sampling Generalized hypergeometric function
Coupon collector's problem Geometric distribution
Keno Notes
^ Rice, John A.. Mathematical Statistics and Data Analysis. Duxbury Press. p. 42.
^ Rivals, I.; Personnaz, L.; Taing, L.; Potier, M.-C. "Enrichment or depletion
of a GO category within a class of genes: which test?". Bioinformatics 23:
401–407. doi:10.1093btl633. PMID 17182697.
^ K. Preacher and N. Briggs. "Calculation for Fisher's Exact Test: An
interactive calculation tool for Fisher's exact probability test for 2 x
2 tables". ursesHandouts/Symmetry%5BPolyaUrn%5D.pdf
References Berkopec, Aleš. "HyperQuick algorithm
for discrete hypergeometric distribution". Journal of Discrete
Algorithms 5: 341. doi:10.1016/j.jda.2006.01.001.
Skala, M.. "Hypergeometric tail inequalities: ending the insanity".
unpublished note External links
The Hypergeometric Distribution and Binomial Approximation to a
Hypergeometric Random Variable by Chris Boucher, Wolfram Demonstrations Project.
Weisstein, Eric W., "Hypergeometric Distribution", MathWorld.