A commitee consists of six men and four women.A randomly chosen subcommittee is made up of three of the committee members.What is the probability that (a) they are all women (b) two of them are men

prob(all women)

= C(4,3)/C(10,3) = ...

prob(two men, 1 woman)
= C(6,2)*C(4,1)/C(10,3) = ....

To find the probability for each scenario, we need to determine the total number of possible subcommittees and the number of subcommittees that meet the given conditions.

Let's start by calculating the total number of possible subcommittees:

The total number of committee members is 6 men + 4 women = 10 members.
To choose a subcommittee of 3 members from this total, we use the combination formula:

C(n, r) = n! / (r!(n-r)!)

C(10, 3) = 10! / (3!(10-3)!)
= 10! / (3!7!)
= (10 * 9 * 8 * 7!) / (3 * 2 * 1 * 7!)
= (10 * 9 * 8) / (3 * 2 * 1)
= 120

So, the total number of possible subcommittees is 120.

(a) Probability that all three subcommittee members are women:

We need to determine the number of subcommittees consisting only of women, which is 4 women out of 10 total members.

C(4, 3) = 4! / (3!(4-3)!)
= 4! / (3!1!)
= 4

Therefore, there are 4 subcommittees consisting only of women.

The probability is given by:
P(all women) = (Number of subcommittees consisting only of women) / (Total number of possible subcommittees)
= 4 / 120
= 1 / 30

So, the probability that all three subcommittee members are women is 1/30.

(b) Probability that two of the subcommittee members are men:

We have 6 men to choose from, and we need to select 2 men out of these.

C(6, 2) = 6! / (2!(6-2)!)
= 6! / (2!4!)
= (6 * 5) / (2 * 1)
= 15

Therefore, there are 15 subcommittees with 2 men and 1 woman.

The probability is given by:
P(2 men) = (Number of subcommittees with 2 men and 1 woman) / (Total number of possible subcommittees)
= 15 / 120
= 1 / 8

So, the probability that two of the subcommittee members are men is 1/8.