Question

Assume that the committee contains 10 men and 5 women and that three are selected at random for a subcommittee.

What is the probability that the subcommittee consists of 2 men and 1 woman, given that it contains both men and women?

Answer 1

The trick to this question is "given that it contains both men and women". So, bear that in mind. Before we start, do you know anything about combinations and permutations?

Answer 2

yes, that's what we've been learning all about in class.

Answer 3

Your eventual denominator will be the total number of ways of selecting a subcommittee with one man together with the total number of ways of selecting a subcommittee with two men because of the "given that ..."

Answer 4

So, your denominator will be P(1 man) + P(2 men). Your numerator will be P(2 men). So, we just find the probability of these two occurences.

Answer 5

Strating with P(1 man), which implies that 2 women are also chosen to make up the three, there are 10 ways to select 1 man, multiplied by the 5 x 4 / 2 ways to select 2 women. Are you with me so far?

Answer 6

you lost me. where do you get the 5 x 4 / 2?

Answer 7

By the way, the way I got 5 x 4 / 2 is 5! / (3! x 2!) . 35 years ago we used the notation 5C2 for 5 combination 2 which equals 10.

Answer 8

5 women. Have to select 2. 5 combination 2. Any of the 5 could be the first one chosen. Leaves 4. 5 x 4. But we have to divide by 2 because order does not matter and we don't want to double-count.

Answer 9

okay.

Answer 10

So, for P(1 man), combining the last 3 posts above, we have 10 x 5C2 or 10 x (5!/(3! x 2!)) or 10 x 10 which = 100. Good so far?

Answer 11

for selecting 2 men of 10 men, we can write = 10C2
for selecting 1 woman of 5 women, we can write = 5C1

Answer 12

So, if you're good on P(1 man), we can go to P(2 men). What do you get for the number of ways to select 2 out of 10, just looking at the men for now. We'll get to the woman in a sec.

Answer 13

45

Answer 14

Very good.

Answer 15

Now, to finish out P(2 men), you have to multiply that 45 by the number of ways to select 1 woman out of 5.

Answer 16

so then what exactly does that 225 mean?

Answer 17

That's just the number of ways to get 2 men and 1 woman, without taking into account the TOTAL number of ways to pick a 3-person subcommittee. It's not a probability yet.

Answer 18

I could have done well to call it N(2 men), not P(2 men).

Answer 19

Out P(2 men and 1 woman) will equal N(2 men) / [ N(2 men) + N(1 man) ]. You have those numbers above.

Answer 20

N(2 men) = 225 and N(1 man) = 100.

Answer 21

That denominator, [ N(2 men) + N(1 man) ], is the total # of ways to have a mixed-gender subcommittee. The numerator, N(2 men), is the total # of ways to have exactly 2 men and 1 woman. So, your probability is this ratio.

Answer 22

So, you have your answer and you're done.

Answer 23

if me, all even = 15C3
so, P(2M, 1W) = (10C2*5C1)/15C3

Answer 24

No, that's not correct RadEn. There is a condition set upon this probability. It's conditional.

Answer 25

RadEn, you can't use 15C3 for the denominator because that would include subcommittees that are all men or all women. Those have to be excluded by the given condition.

Answer 26

thank you, @tcarroll010 !

Answer 27

You're very welcome! Good luck to you and all your studies.