Question

A delegation of 3 is selected from a city council made up of 5 liberals and 4 conservatives. What is the expected number of liberals in the delegation? What is the expected number of conservatives?

Answer 1

To calculate the expected number of liberals and conservatives in a 3-person delegation from a council of 5 liberals and 4 conservatives, the concept of expected value is used, involving probability calculations for each possible number of liberals or conservatives and summing the products of these probabilities by the number of liberals or conservatives.

Step-by-step explanation:

The student's question pertains to the expected number of liberals and conservatives when selecting a delegation of 3 from a city council made up of 5 liberals and 4 conservatives. To solve this, we use the concept of expected value, which in a probability context is the average outcome if the experiment were repeated many times.

Let's define the random variable X as the number of liberals in the delegation. The probabilities for X=0,1,2,3 are obtained by considering all possible ways to choose the delegation and the ways that include the specified number of liberals. For instance, the probability of having 3 liberals (X=3) in the delegation is the number of ways to choose 3 liberals from 5 divided by the number of ways to choose 3 delegates from 9 total members.

Expected value E[X] for the number of liberals is calculated by the sum of each outcome multiplied by its probability. Similarly, the random variable Y for the number of conservatives follows the same process.

Step-by-Step Explanation:

1. Calculate the probability for each number of liberals (X=0,1,2,3) and conservatives (Y=0,1,2,3) that can be chosen in the delegation.
2. Multiply these probabilities by the number of liberals or conservatives accordingly to contribute to the expected value.
3. Sum these products to find the expected values E[X] for liberals and E[Y] for conservatives.

To obtain the final expected value for liberals and conservatives, the following specific calculations are needed: for X=0,1,2, and 3, determine the probabilities and multiply by 0, 1, 2, and 3 respectively, and then sum these. Reverse the roles of liberals and conservatives to calculate E[Y].

The expected number of liberals E[X] would therefore take into account the combination of 5 choose 3, 5 choose 2, and so on, while the expected number of conservatives E[Y] involves similar calculations with the 4 conservatives.