Final answer:
To calculate the joint probability of X ≤ 3 and Y > 3, multiply the probabilities of each event together. To find the expected value of X * Y, multiply each outcome by its probability and sum the results. To find the expected value of 1 / Y, multiply each outcome by its probability and sum the results.
Step-by-step explanation:
a) To calculate the joint probability of X ≤ 3 and Y > 3, we need to determine the probability of each event separately. The probability of X ≤ 3 is the probability of rolling a 1, 2, or 3, which is 3/6 or 1/2. The probability of Y > 3 is the probability of rolling a 4, 5, or 6, which is also 3/6 or 1/2. To find the joint probability, we multiply these probabilities together: 1/2 * 1/2 = 1/4.
b) The expected value of X * Y is found by multiplying each possible outcome of X * Y by its probability and summing the results. The possible outcomes of X * Y are: 1, 2, 3, 4, 5, 6, 8, 9, 12. The probabilities of each outcome can be found by multiplying the probability of X with the probability of Y for each pair of values. For example, the probability of X = 1 and Y = 1 is 1/6 * 1/6 = 1/36. After calculating the probabilities for each outcome, multiply each outcome by its probability and sum the results to find the expected value.
c) The expected value of 1 / Y is found by multiplying each possible outcome of 1 / Y by its probability and summing the results. The possible outcomes of 1 / Y are: 1/6, 1/5, 1/4, 1/3, 1/2, 1. The probabilities of each outcome can be found by multiplying the probability of Y with the corresponding value. For example, the probability of Y = 1 is 1/6, so the probability of 1 / Y = 1 is also 1/6. After calculating the probabilities for each outcome, multiply each outcome by its probability and sum the results to find the expected value.