Binomial experiments require that any sampling be done with replacement because each trial must be independent of the others. The hypergeometric distribution also has two outcomes: success and failure. The sampling, however, is done without replacement. For a population of N items having k successes and N - k failures, the probability of selecting a sample of size n that has x successes and n – x failures is given by
P(x) = (kCx) (N-kCn-x) / NCn
In a shipment of 15 microchips, 2 are defective and 13 are not defective. A sample of three microchips is chosen at random. Find the probability that
(a) All three microchips are not defective,
(b) One microchip is defective and two are not defective,
(c) Two microchips are defective and one is not defective.