Assume you have a typical fair 6-sided die. That is to say, it has sides 1, 2, 3, 4, 5, 6 and each side has probability of 1/6 of occurring. You are to roll the die twice, and record the maximum, M, of the two 6 rolls. If you roll a 2 and a 3 (or a 3 and a 2), then M = 3. If you roll a 4 and a 4, then M = 4. The rolls are independent of each other.

a. Derive the probability table for M. That is to say P(M = m) for m=1,2,3,4,5,6. b. Find the expected value of M, E(M). c. What is the variance of M, var(M). d. Given that the maximum rolled is a 4, what is the probability that one of the rolls was a 2 ?