In probability theory and statistics, the hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement.

There is a shipment of M objects in which K are defective. The hypergeometric distribution describes the probability that in a sample of n distinctive objects drawn from the shipment exactly x objects are defective.

In general, if a random variable X follows the hypergeometric distribution with parameters M, K and n, then the probability of getting exactly x successes is given by

f(x,M,K,n) = over(K,x)over(M-k,n-x)/over(M,n)

where  over(n,k) = n!/(k!(n-k)!)

The formula can be understood as follows: There are over(M,n) possible samples (without replacement). There are over(K,x) ways to obtain x defective objects and there are over(M-k,n-x) ways to fill out the rest of the sample with non-defective objects.

The fact that the sum of the probabilities, as x runs through the range of possible values, is equal to 1, is essentially Vandermonde’s identity from combinatorics.

support max(0,K+n-M) ⇐ x ⇐ min(K,n)