From a group of 10 people, 5 men and 5 women, six are to be chosen to serve on a committee.
a) In how many ways can the committee be chosen?
b) Find the probability that two men will be chosen.
c) Find the probability that the committee chosen will consist at least 4 women.
number of ways to choose 6 from 10
= C(10,6) = 1056
b) number with 2 men and 4 women
= C(5,2) x C(5,4)
= 10(4) = 40
prob(choosing 2men and 4 women) = 40/1056
= 5/132
c) at least 4 women --> 4 women or 5 women
= C(5,4)C(5,2) + C(5,5)C(5,1)
= 5(40) + 1(5)
= 205
prob(at least 4 women) = 205/1056
the committee can be chosen in these different ways
4b 2g and it6 can also be
4g 2b in many other ways!!!
6b 0g
6g ob
3b 3g
5g 1b
5b 1b
if what i posted wasn't your question sorry cause after i submited that i started thinking that it wasnt..........
To solve this problem, we need to use combinations.
a) In order to find the number of ways the committee can be chosen, we need to determine the number of combinations of 6 people that can be selected from a group of 10. This can be calculated using the formula for combinations:
nCr = (n!) / (r!(n-r)!)
where n is the total number of items and r is the number of items to be selected.
In this case, we have 10 people and we need to choose 6, so the number of ways the committee can be chosen is:
10C6 = (10!) / (6!(10-6)!) = 210
Therefore, there are 210 different ways to form the committee.
b) To find the probability that exactly 2 men will be chosen, we need to determine the number of ways this can happen and divide it by the total number of ways the committee can be chosen.
First, let's calculate the number of ways to choose 2 men from a group of 5:
5C2 = (5!) / (2!(5-2)!) = 10
Next, we need to choose the remaining 4 members from the remaining pool of 8 people (3 men and 5 women):
8C4 = (8!) / (4!(8-4)!) = 70
Now, we multiply the number of ways to choose 2 men by the number of ways to choose 4 additional members:
10 * 70 = 700
To find the probability, we divide this number by the total number of ways the committee can be chosen:
Prob(2 men) = 700 / 210 = 10/3 or approximately 3.33
Therefore, the probability that exactly 2 men will be chosen is approximately 3.33.
c) To find the probability that the committee will consist of at least 4 women, we need to calculate the number of ways to choose 4 or 5 women and divide it by the total number of ways the committee can be chosen.
First, let's calculate the number of ways to choose exactly 4 women:
5C4 = (5!) / (4!(5-4)!) = 5
Next, let's calculate the number of ways to choose exactly 5 women:
5C5 = (5!) / (5!(5-5)!) = 1
Now, let's multiply these numbers by the number of ways to choose the remaining members (2 or 1, respectively):
5 * 6C2 = 5 * (6!) / (2!(6-2)!) = 60
1 * 6C1 = 1 * (6!) / (1!(6-1)!) = 6
Now, we add these two numbers together to get the total number of ways to choose at least 4 women:
60 + 6 = 66
To find the probability, we divide this number by the total number of ways the committee can be chosen:
Prob(at least 4 women) = 66 / 210 ≈ 0.3143
Therefore, the probability that the committee will consist of at least 4 women is approximately 0.3143.