Probability Question

Question

A company has 27 employes. A committee of 4 will be selected. The probability that exactly one woman is selected for the committee is two times the probability that no woman are selected. Calculate the probability that exactly 2 woman are selected.

Answer 1

This is one of the best problems I have ever seen presented here. Assume that there are n women (and thus 27-n men.

The stated probability that exactly one woman is selected for the committee is two times the probability that no woman are selected means P(1 woman, 3 men) = 2•P(4 men), i.e.

 

nC1•(27-n)C3 / 27C4 = 2•(27-n)C4 / 27C4.

 

Since the denominators of the above equality are identical, they can be dropped ("clearing fractions").

Then we have n•(27-n)! / [3! (24-n)!] = 2•(27-n)! / [(23-n)! 4!] and the identical numerator factors (23-n)! can be divided out, so that we have simplified to

 n•1/[3!(24-n)!] = 2•1/[(23-n)!4!]

 

With a little more simplification we wind up with 12n = 6(24-n), leading to n = 8.

 

Then P(2 women, 2 men) = 8C2•19C2 / 27C4 ≈ 0.2728