A committee of 3 members is selected from a club made up of 10 junior members and 20 senior members. What is the expected number of juniors in the committee?
6728
1140
563
To find the expected number of juniors in the committee, we need to determine the probability of each possible outcome and then calculate the weighted average.
First, let's calculate the probability of selecting a certain number of junior members in the committee. There are a total of 30 members, and we need to select 3 members for the committee.
The number of ways to choose a specific number of junior members is calculated using combinations. The formula for combinations is:
C(n, k) = n! / (k!(n-k)!)
where n is the total number of members (30), and k is the number of junior members to be chosen.
Let's calculate the probabilities for each possible outcome:
For selecting 0 junior members:
C(10, 0) = 10! / (0!(10-0)!) = 1
The probability of selecting 0 junior members is 1.
For selecting 1 junior member:
C(10, 1) = 10! / (1!(10-1)!) = 10
The probability of selecting 1 junior member is 10.
For selecting 2 junior members:
C(10, 2) = 10! / (2!(10-2)!) = 45
The probability of selecting 2 junior members is 45.
For selecting 3 junior members:
C(10, 3) = 10! / (3!(10-3)!) = 120
The probability of selecting 3 junior members is 120.
Next, we calculate the expected number of juniors in the committee by taking the weighted average:
Expected number of juniors = (0 * Probability of 0 juniors) + (1 * Probability of 1 junior) + (2 * Probability of 2 juniors) + (3 * Probability of 3 juniors)
Expected number of juniors = (0 * 1) + (1 * 10) + (2 * 45) + (3 * 120) = 0 + 10 + 90 + 360 = 460
Therefore, the expected number of juniors in the committee is 460.