probability question

Question

Two people are selected to serve as executives from a committee of 3 men and 3 women. In a random selection for the two executives, what is the probability that the selection results in one man and one woman? Please show all of your work

Answer 1

The sample space, ie all of the possible outcomes, is comprised of 6C2 distinct, 2-person committees. 6C2 = 15, which will be the denominator of our final probability calculation.

Of those 15, we have 3C1 · 3C1 distinct committees with exactly one man and one woman. 3C1·3C1 = 9

9/15 = 3/5 = 60% This jibes with our intuition that this would be a fairly likely outcome, more likely than a homogeneous committee.

We can arrive at the same answer applying this logic: Regardless of the gender of the first choice of committee member, you will have a 3/5 chance of picking someone of the opposite gender to serve.