Problem Solving

If a jury of 12 people is to be selected randomly from a pool of 15 potential jurors, and the jury pool consists of 2/3 men and 1/3 women, what is the probability that the jury will comprise at least 2/3 men?



A)24/91


B)45/91


C)2/3


D)67/91


E)84/91

OAD

j_shreyans wrote:If a jury of 12 people is to be selected randomly from a pool of 15 potential jurors, and the jury pool consists of 2/3 men and 1/3 women, what is the probability that the jury will comprise at least 2/3 men?

A) 24/91
B) 45/91
C) 2/3
D) 67/91
E) 84/91

Question rephrased: What is the probability that at least 8 men will be selected to serve on the 12-member jury?

P(good outcome) = 1 - P(bad outcome).

Here, a BAD outcome means selecting a jury with FEWER than 8 men.
Of the 10 men and 5 women in the jury pool, 3 people must be selected NOT to serve on the jury.
There is only ONE WAY to select fewer than 8 men FOR the jury:
The 3 people selected NOT to serve on the jury must ALL be men, leaving 7 men and all 5 women to serve on the jury.

P(1st non-juror is a man) = 10/15. (Of the 15 people in the jury pool, 10 are men.)
P(2nd non-juror is a man) = 9/14. (Of the 14 remaining people in the jury pool, 9 are men.)
P(3rd non-juror is a man) = 8/13. (Of the 13 remaining people in the jury pool, 8 are men.)
Since a bad outcome requires that all 3 events happen, we MULTIPLY the fractions:
10/15 * 9/14 * 8/13 = 24/91.

Thus:
P(good outcome) = 1 - 24/91 = 67/91.

The correct answer is D.

Hi GMATGuru,

Cant we attempt this question by selecting like:
Probability of selecting (8M & 4W) + Prob. (9M & 3W) + Prob. (10M & 2W)!!

Please explain!
Thanks.

GMATGuruNY wrote:

j_shreyans wrote:If a jury of 12 people is to be selected randomly from a pool of 15 potential jurors, and the jury pool consists of 2/3 men and 1/3 women, what is the probability that the jury will comprise at least 2/3 men?

A) 24/91
B) 45/91
C) 2/3
D) 67/91
E) 84/91

Question rephrased: What is the probability that at least 8 men will be selected to serve on the 12-member jury?

P(good outcome) = 1 - P(bad outcome).

Here, a BAD outcome means selecting a jury with FEWER than 8 men.
Of the 10 men and 5 women in the jury pool, 3 people must be selected NOT to serve on the jury.
There is only ONE WAY to select fewer than 8 men FOR the jury:
The 3 people selected NOT to serve on the jury must ALL be men, leaving 7 men and all 5 women to serve on the jury.

P(1st non-juror is a man) = 10/15. (Of the 15 people in the jury pool, 10 are men.)
P(2nd non-juror is a man) = 9/14. (Of the 14 remaining people in the jury pool, 9 are men.)
P(3rd non-juror is a man) = 8/13. (Of the 13 remaining people in the jury pool, 8 are men.)
Since a bad outcome requires that all 3 events happen, we MULTIPLY the fractions:
10/15 * 9/14 * 8/13 = 24/91.

Thus:
P(good outcome) = 1 - 24/91 = 67/91.

The correct answer is D.