ABSTRACT
Consider a lot consisting of N items of which M of them are defective and the remaining N −M of them are nondefective. A sample of n items is drawn randomly without replacement. (That is, an item sampled is not replaced before selecting another item.) Let X denote the number of defective items that is observed in the sample. The random variable X is referred to as the hypergeometric random variable with parameters N and M . Noting that the number of ways one can select b different objects from a collection of a different objects is(
a
b
) =
a! b!(a− b)! ,
we find that the number of ways of selecting k defective items from M defective items is
) ; the number of ways of selecting n − k nondefective items from
N−M nondefective items is (N−Mn−k ). Therefore, total number of ways of selecting n items with k defective and n−k nondefective items is (Mk )(N−Mn−k ). Finally, the number of ways one can select n different items from a collection of N different items is
) . Thus, the probability of observing k defective items in a sample of
n items is given by
f(k|n,M,N) = P (X = k|n,M,N) = (M k
) , L ≤ k ≤ U, where L = max{0,M −N + n} and U = min{n,M}.
