The hypergeometric distribution describes the probability of r "successful" outcomes occurring from n experiments, where the elements drawn are not returned to the population, i.e. do not participate in the next trial.
The probability mass function of the hypergeometric distribution is:
The main use of the hypergeometric distribution is in sampling without replacement.
Example: Hypergeometric Distribution
Let's assume that there are 50 balls in a hat, 5 of which are white and 45 black ones. With your eyes closed, 10 balls are drawn from the hat, which are not returned to the hat once they are drawn. What is the probability that 4 of the 10 balls drawn will be white?
Answer: If we insert the known values into the above formula, we get
P(X=4)=(5!/(4!*1!)*45!/(6!*39!))/(50!/(10!*40!))=0.003965
Based on the data from the example, the diagram shows the probabilities for the values of the random variable from 0 to 5.
The expected value of a random variable that has a hypergeometric distribution is calculated using the formula
The formula for calculating the variance of a random variable that has a hypergeometric distribution is
If n = 1, then the hypergeometric distribution and the Bernoulli distribution are equal.
If the population size N is extremely large relative to the sample size n (typically n/N ≤ 0.05), the hypergeometric distribution can be approximated by the binomial distribution.
The difference between the hypergeometric distribution and the binomial distribution is that the condition for the existence of the binomial distribution is the possibility of re-selection of already selected elements of the set from which it is selected, while re-selection is not allowed in the hypergeometric distribution.
binomial distribution
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