A committee of 4 person is to be appointed from 7 men and 3 women, what is the probability that the committee contain exactly 2 women. At least one man.
Good
To find the probability that the committee contains exactly 2 women and at least one man, we need to find the total number of possible committees with these criteria and divide it by the total number of possible committees.
First, let's find the total number of possible committees. We have a total of 7 men and 3 women, which means we have 10 people in total. We need to choose 4 people for the committee. This can be done using the combination formula, which is given by:
C(n, r) = n! / (r!(n-r)!)
In this case, n = 10 (total people) and r = 4 (committee size).
So, the total number of possible committees is:
C(10, 4) = 10! / (4!(10-4)!) = 10! / (4!6!) = (10*9*8*7) / (4*3*2*1) = 210
Now, let's find the number of committees that contain exactly 2 women. We have 3 women to choose from, and we need to select 2 of them. This can be done using the combination formula:
C(3, 2) = 3! / (2!(3-2)!) = 3! / (2!1!) = (3*2*1) / (2*1) = 3
Next, we need to choose the remaining 2 people from the remaining 7 men. This can be done using the combination formula:
C(7, 2) = 7! / (2!(7-2)!) = 7! / (2!5!) = (7*6) / (2*1) = 21
Therefore, the number of committees with exactly 2 women is:
C(3, 2) * C(7, 2) = 3 * 21 = 63
Finally, let's find the number of committees that contain at least one man. To do this, we'll find the total number of possible committees with only women (0 men) and subtract it from the total number of possible committees.
The number of committees with only women is given by:
C(3, 4) = 3! / (4!(3-4)!) = 3! / (4!(-1)!) = 0
Therefore, the number of committees with at least one man is:
Total number of committees - Number of committees with only women = 210 - 0 = 210
So, the probability that the committee contains exactly 2 women and at least one man is:
(Number of committees with exactly 2 women and at least one man) / (Total number of committees) = (63 / 210) ≈ 0.3 or 30%
To find the probability that the committee contains exactly 2 women and at least one man, we need to calculate the total number of possible committees and the number of committees that fulfill these criteria.
Step 1: Calculating the total number of possible committees:
The total number of possible committees is calculated using the combination formula. In this case, we need to choose 4 people from a pool of 7 men and 3 women. The formula for combinations is given by:
nCr = n! / (r! * (n-r)!)
where n is the total number of people and r is the number of people to be chosen.
In our case, n = 10 (7 men + 3 women) and r = 4 (the committee size).
So, the total number of possible committees is:
10C4 = 10! / (4! * (10-4)!) = (10 * 9 * 8 * 7) / (4 * 3 * 2 * 1) = 210
Step 2: Calculating the number of committees with exactly 2 women and at least one man:
To find the number of committees with exactly 2 women and at least one man, we can use the concept of complementary probability. We first calculate the number of committees with no man, no woman, or only 1 woman, and then subtract it from the total number of committees.
Number of committees with no man:
Since we need at least one man, we cannot have a committee with no man.
Number of committees with no woman:
Since we need exactly 2 women, we cannot have a committee with no woman.
Number of committees with only 1 woman:
We can choose 1 woman in 3 ways (since there are 3 women available) and then choose the remaining 3 people from the 7 men. So, the number of such committees is:
3C1 * 7C3 = 3 * (7! / (3! * (7-3)!)) = 3 * (7 * 6 * 5) / (3 * 2 * 1) = 35
So, the number of committees with exactly 2 women and at least one man is:
Total number of committees - (number of committees with no man + number of committees with no woman + number of committees with only 1 woman)
= 210 - (0 + 0 + 35) = 210 - 35 = 175
Step 3: Calculating the probability:
Finally, we can calculate the probability by dividing the number of committees with exactly 2 women and at least one man by the total number of possible committees.
Probability = Number of committees with exactly 2 women and at least one man / Total number of possible committees
= 175 / 210
= 5/6
Therefore, the probability that the committee contains exactly 2 women and at least one man is 5/6 or approximately 0.8333.