In probability statistics, the hypergeometric distribution is a discrete probability distribution that describes the probability of k successes in n draws, without replacement, from a finite population of size N that contains exactly K objects with that feature, wherein each draw is either a success or a failure.
Mathematical definition
Properties of Hypergeometric Distribution
 The distribution tends to binomial distribution if N➝∞ and K/N⟶p.
 Hypergeometric distribution is symmetric if p=1/2; positively skewed if p<1/2; negatively skewed if p>1/2.
 The mean of the hypergeometric distribution coincides with the mean of the binomial distribution if M/N=p.
Mean 

Variance 

CF 

MGF 


Application of Hypergeometric Distribution
 The distribution is commonly studied in most introductory probability courses.
 In introducing students to distribution, drawing balls from an urn or selecting playing cards from a deck of cards are often discussed. This is a simple process which focus on sampling without replacement.
 We are also used the distribution to estimate the number of fishes in a lake.
 Election audits typically test a sample of machinecounted precincts to see if recounts by hand or machine match the original counts.
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 k successes (out of n total draws) from a population of size N containing K successes. In a test for overrepresentation of successes in the sample, the hypergeometric pvalue is calculated as the probability of randomly drawing k or more successes from the population in n total draws. In a test for underrepresentation, the pvalue is the probability of randomly drawing k or fewer successes.
The test based on the hypergeometric distribution is identical to the corresponding onetailed version of Fisher’s exact test. ^{}Reciprocally, the pvalue of a twosided Fisher’s exact test can be calculated as the sum of two appropriate hypergeometric tests (for more information see^{}). The test is often used to identify which subpopulations are over or underrepresented 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 overrepresentation of various demographic subgroups (e.g., women, people under 30).