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Hypergeometric Distribution


The hypergeometric distribution is a discrete probability distribution that arises when we try to draw a random sample without replacement from a given population. For example, suppose there are N balls in a bag out of which M are white and the remaining N-M are black. Suppose we choose a sample of size n from the bag. Then the probability that the sample has k white balls can be calculated using the pdf of the hypergeometric distribution.

Hypergeometric probability distribution
Hypergeometric distribution pdf formula

Example: Suppose that a bag contains 10 balls out of which 6 are white and 4 are black. If we choose 4 balls randomly from the bad then calculate the probability that two of these balls are white.

Solution: Let X denote the number of white balls in our sample.

Given N=10, M=6, n=4, and k=2.

Substituting all this in the above formula we get,

P(X=2) = (62)*(42)/(104) = 15*6/210 = 90/210 = 0.429

Mean and Variance of Hypergeometric Distribution:

Mean and variance of hypergeometric distribution
Hyper geometric distribution – Mean and Variance

Example: In the above example, calculate the mean number of white balls selected and the variance.

Solution: Given N=10, M=6, n=4 and k=2.

Substituting all this in the above formula we get,

Mean = 4*6/10 = 2.4 ,that is, 2.4 white balls will be chosen on average.

Variance = (10*6*4*6)/(100*9) = 1.6

Use of hypergeometric distribution:

  1. It is used in the theory of Quality control to calculate the number of lots to be selected for inspection of quality of goods.
  2. It is used in sampling theory to calculate the probabilities of selecting particular kinds of samples without replacement.

Binomial approximation to Hypergeometric Distribution:

As the value of N tends toward infinity and the value M/N approaches a finite number then the hypergeometric distribution can be approximated using the pdf of the binomial distribution with parameters n and p=M/N.

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