If a jury of 12 people is to be selected randomly from a

[BTGmoderatorDC]

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

OA D

Source: Manhattan Prep

[GMATGuruNY]

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.

[Scott@TargetTestPrep]

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

OA D

Source: Manhattan Prep

We see that 2/3 x 15 = 10 potential jurors are male and thus 5 are female.

We want to have at least 2/3 x 12 = 8 men to be selected as jurors and thus at most 4 women to be selected as jurors.

The number of ways to select 8 men and 4 women is 10C8 x 5C4 = 10C2 x 5C1 = (10 x 9)/2 x 5 = 45 x 5 = 225.

The number of ways to select 9 men and 3 women is 10C9 x 5C3 = 10C1 x 5C2 = 10 x (5 x 4)/2 = 10 x 10 = 100.

The number of ways to select 10 men and 2 women is 10C10 x 5C2 = 1 x (5 x 4)/2 = 1 x 10 = 10.

The total number of ways of selecting 12 people from 15 is 15C12 = 15C3 = (15 x 14 x 13)/(3 x 2) = 5 x 7 x 13 = 455.

Therefore, the probability that the jury will comprise at least 2/3 men is (225 + 100 + 10)/455 = 335/455 = 67/91.

Answer: D