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.

**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) = (^{6}_{2})*(^{4}_{2})/(^{10}_{4}) = 15*6/210 = 90/210 = 0.429

**Mean and Variance of Hypergeometric Distribution:**

**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**:

- It is used in the theory of Quality control to calculate the number of lots to be selected for inspection of quality of goods.
- 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.