Final answer:
The expected number of liberals in a delegation of 3 selected from a city council of 3 liberals and 3 conservatives is 1.5.
Step-by-step explanation:
The question pertains to determining the expected number of liberals in a delegation of 3 selected from a council with an equal number of liberals and conservatives. To calculate this expectation, we consider all possible selections of the delegation and their respective probabilities.
We can have the following combinations: 3 liberals, 2 liberals and 1 conservative, 1 liberal and 2 conservatives, or 3 conservatives. The probability of each combination can be calculated using combinations, and the expected number of liberals is the sum of the number of liberals in each combination multiplied by the respective probability of that combination occurring.
Thus, we calculate the expectation E(X) as follows:
- P(3 liberals) = (3 choose 3) * (3 choose 0) / (6 choose 3) = 1/20
- P(2 liberals, 1 conservative) = (3 choose 2) * (3 choose 1) / (6 choose 3) = 3/10
- P(1 liberal, 2 conservatives) = (3 choose 1) * (3 choose 2) / (6 choose 3) = 3/10
- P(0 liberals) = (3 choose 0) * (3 choose 3) / (6 choose 3) = 1/20
E(X) = 3*(1/20) + 2*(3/10) + 1*(3/10) + 0*(1/20) = 1.5
Hence, the expected number of liberals in the delegation is 1.5.