From a group of 3 men and 4 women, a delegation of 2 is selected at random. What is the expected number of men in the delegation? Round your answers to two decimal places as needed.
Could someone help me with how to do this?
Pr(zero men)=4/7*3/6=12/42
pr (one man)=3/7*4/6+4/7*3/6=24/42
Pr(two men)= 3/7*2/6= 6/42
Now if you mean by "expected" value, then one man is expected.
Of course, I'd be happy to help you with this problem! To find the expected number of men in the delegation, we need to determine the probability of selecting each possible combination of two individuals and the number of men in each combination.
There are a total of C(7, 2) = 21 possible combinations when selecting 2 individuals from a group of 7 (3 men + 4 women). Let's calculate the probability of each combination:
1. Probability of selecting 2 men (MM): C(3, 2) / C(7, 2) = 3/21 = 1/7
2. Probability of selecting 1 man and 1 woman (MW or WM): C(3, 1) * C(4, 1) / C(7, 2) = (3 * 4) / 21 = 12/21 = 4/7
3. Probability of selecting 2 women (WW): C(4, 2) / C(7, 2) = 6/21 = 2/7
Now, let's consider the number of men in each combination:
1. MM (2 men) → Number of men = 2
2. MW or WM (1 man and 1 woman) → Number of men = 1
3. WW (2 women) → Number of men = 0
Finally, we multiply the probability of each combination by the corresponding number of men, and sum them up:
Expected number of men = (Probability of MM * Number of men in MM) + (Probability of MW or WM * Number of men in MW or WM) + (Probability of WW * Number of men in WW)
Expected number of men = (1/7 * 2) + (4/7 * 1) + (2/7 * 0) = 2/7 + 4/7 + 0 = 6/7
Therefore, the expected number of men in the delegation is approximately 0.86 when rounded to two decimal places.
I hope this helps! If you have any more questions, feel free to ask.
To find the expected number of men in the delegation, we need to calculate the probability of each possible outcome and multiply it by the number of men for that outcome.
First, let's calculate the total number of possible delegations. We have 3 men and 4 women, so the total number of delegations is given by the combination formula:
Total delegations = C(7, 2) = 7! / (2!(7-2)!) = 21
Now we can calculate the probability of each outcome:
1. Selecting 2 men:
The number of ways to select 2 men out of 3 is given by the combination formula: C(3, 2) = 3! / (2!(3-2)!) = 3
So, the probability of selecting 2 men is P(1) = 3/21 = 1/7.
2. Selecting 1 man and 1 woman:
The number of ways to select 1 man out of 3 is C(3, 1) = 3! / (1!(3-1)!) = 3. The number of ways to select 1 woman out of 4 is C(4, 1) = 4! / (1!(4-1)!) = 4.
The total number of combinations of selecting 1 man and 1 woman is 3 * 4 = 12.
So, the probability of selecting 1 man and 1 woman is P(2) = 12/21 = 4/7.
3. Selecting 2 women:
The number of ways to select 2 women out of 4 is C(4, 2) = 4! / (2!(4-2)!) = 6.
So, the probability of selecting 2 women is P(3) = 6/21 = 2/7.
Finally, we multiply the probability of each outcome by the number of men in that outcome and sum them up to find the expected number of men:
Expected number of men = P(1) * 2 + P(2) * 1 + P(3) * 0
= (1/7) * 2 + (4/7) * 1 + (2/7) * 0
= 2/7 + 4/7 + 0
= 6/7
Rounding to two decimal places, the expected number of men in the delegation is approximately 0.86.
To find the expected number of men in the delegation, we need to calculate the probability of selecting each possible combination of men and women, and then multiply it by the number of men in that combination. Finally, we sum up all the products to get the expected number.
Here's how we can do it step by step:
1. Calculate the total number of ways to choose a delegation of 2 from a group of 3 men and 4 women. This can be done using combinations:
C(n, r) = n! / (r! * (n - r)!)
where n is the total number of elements and r is the number of elements to be chosen.
In this case, we have 7 people in total (3 men + 4 women) and we want to choose a delegation of 2. So,
C(7, 2) = 7! / (2! * (7 - 2)!) = 21
2. Calculate the number of ways to choose a delegation with 0, 1, and 2 men. This can be done by counting the number of ways to choose men and women separately.
- To choose a delegation with 0 men, we need to choose 2 women:
C(4, 2) = 4! / (2! * (4 - 2)!) = 6
- To choose a delegation with 1 man, we can choose 1 man from the 3 available and 1 woman from the 4 available:
C(3, 1) * C(4, 1) = (3! / (1! * (3 - 1)!)) * (4! / (1! * (4 - 1)!)) = 3 * 4 = 12
- To choose a delegation with 2 men, we need to choose 2 men:
C(3, 2) = 3! / (2! * (3 - 2)!) = 3
3. Calculate the probability of each combination by dividing the number of ways to choose each combination by the total number of possible combinations:
- For a delegation with 0 men:
6 / 21 = 2 / 7
- For a delegation with 1 man:
12 / 21 = 4 / 7
- For a delegation with 2 men:
3 / 21 = 1 / 7
4. Finally, calculate the expected number of men by multiplying the number of men in each combination by the probability of that combination.
- For a delegation with 0 men:
0 * (2 / 7) = 0
- For a delegation with 1 man:
1 * (4 / 7) = 4 / 7 ≈ 0.57
- For a delegation with 2 men:
2 * (1 / 7) = 2 / 7 ≈ 0.29
5. Sum up the products to get the expected number of men in the delegation:
0 + 4/7 + 2/7 = 6/7 ≈ 0.86
Therefore, the expected number of men in the delegation is approximately 0.86 (rounded to two decimal places).