Two fair eight-sided dice have their faces numbered from 1 to 8. What is the expected value of the sum of the rolls of both dice?
I know that the expected value of a random variable is the average of its possible outcomes, weighted by their probabilities. I also know that the expected value of the sum of two independent random variables is the sum of their expected values. So I can find the expected value of the sum of the rolls by adding the expected values of each roll. To find the expected value of one roll, I need to consider all the possible outcomes and their probabilities. Since the dice are fair, each outcome has the same probability of 1/8. The possible outcomes are 1, 2, 3, 4, 5, 6, 7, and 8. To find the average of these outcomes, I need to add them up and divide by the number of outcomes. The sum of the numbers from 1 to 8 is (8 * 9) / 2 = 36, by a formula for the sum of an arithmetic series. The number of outcomes is 8. So the average of the outcomes is 36 / 8 = 4.5. This means that the expected value of one roll is 4.5. Therefore, the expected value of the sum of the rolls is 4.5 + 4.5 = 9.

