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?
To find the expected number of men in the delegation, we need to calculate the probability of selecting each possible outcome and multiply it by the number of men in that outcome.
First, let's calculate the probability of selecting different outcomes:
1. Selecting 0 men and 2 women:
- The probability of selecting a woman on the first pick is 4/7.
- The probability of selecting another woman on the second pick is 3/6 (since there are now 3 women left out of 6 remaining people).
- The probability of this outcome is (4/7) * (3/6) = 2/7.
2. Selecting 1 man and 1 woman:
- The probability of selecting a man on the first pick is 3/7.
- The probability of selecting a woman on the second pick is 4/6 (since there are 4 women left out of 6 remaining people).
- The probability of this outcome is (3/7) * (4/6) = 2/7.
3. Selecting 2 men:
- The probability of selecting a man on the first pick is 3/7.
- The probability of selecting another man on the second pick is 2/6 (since there are now 2 men left out of 6 remaining people).
- The probability of this outcome is (3/7) * (2/6) = 1/7.
Now, let's calculate the expected number of men:
Expected number of men = (Number of men in 0 men and 2 women outcome * Probability of 0 men and 2 women) +
(Number of men in 1 man and 1 woman outcome * Probability of 1 man and 1 woman) +
(Number of men in 2 men outcome * Probability of 2 men)
Expected number of men = (0 * 2/7) + (1 * 2/7) + (2 * 1/7)
Expected number of men = 0 + 2/7 + 2/7
Expected number of men = 4/7
So, the expected number of men in the delegation is 4/7, which is approximately 0.57 (rounded to two decimal places).
0: p(0) ((3C0)(4C2)/7C2)
1: p(1) ((3C1)(4C1)/7C2))
2: p(2) ((3C2)(4C0)/7C2))
Now to find expected number, multiply the p by the probability. Good luck!