Question

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

Answer 1

D.
`(67)/(91)`

Explanation:

Given,

Total number of jurors = 15,

Out of which 2/3 are men and 1/3 are women,

So, total men =
`(2)/(3)* 15` = 10,

And, total women = 15 - 10 = 5,

The people have to select = 12,

So, the total ways of selecting 12 people =
`^(15)C_(12)=(15!)/(12!* 3!)=455`

Now, 2/3 of 12 =
`(2)/(3)* 12` = 2 × 4 = 8,

Thus, there would be at least 8 men.

Now, the possible ways of selecting at least 8 men = 8 men 4 women + 9 men 3 women + 10 men 2 women

`=^(10)C_8* ^5C_4 + ^(10)C_9* ^5C_3+^(10)C_(10)* ^5C_2`

`=(10!)/(8!* 2!)* (5!)/(4!* 1!)+(10!)/(9!* 1!)* (5!)/(3!* 2!)+(10!)/(10!* 0!)* (5!)/(2!* 3!)`

= 225 + 100 + 10

= 335

Hence, the probability that the jury will comprise at least 2/3 men

`=\frac{\text{the possible ways of selecting at least 8 men}}{\text{total ways of selecting 12 people}}`

`=(335)/(455)`

`=(67)/(91)`

OPTION D is correct.