A club has 21 members, four of whom are executives. Five members are randomly selected to take part in a survey. What is the probability that four non-executive members are chosen?

Answer 1

On problems like this, we can either explicitly provide the combinations in the numerator and denominator or plug into the hypergeometric distribution on your calculator/Excel/...

Explicitly stating the combinations, as there are 21 members and 4 executives, there are 17 non-executives.

Then, selecting 4 non-executives of 17 and 1 executive of 4 when selecting 5 from 21 to survey will be

C(17,4)C(4,1)/C(21,5) = 2380 * 4/20349 = 80/171 = .4678

(The old-fashioned way to arrive at 80/171 would be to explicitly write out the combinations and factor.)

With the hypergeometric distribution, we could simply write = hypgeom.dist(4,5,17,21,0) - the final 0 is because we are looking for exactly 4, as we are finding 4 of 5 non-executives in the sample when there are 17 of 21 in the population.