Problem 3.9 (rolling dice, part 2). Suppose that we roll two distinct n-sided, fair dice (let's call them Die 1 and Die 2): the faces of each die are marked 1, 2, ..., n, and they are equally likely. Here, the number n is generic and your answers must be given in terms of n. (a) What is the sample space S, and what is its size #(S)? (b) What is the probability of the event E1 = {(1, 2) }, i.e. that the first die shows 1 and the second shows 2? (c) What is the probability that both dice show the same number? In other words, what is the probability of the event E2 = {(1, 1), (2, 2), ..., (n, n) }? (d) This is tricky. What is the probability that the number shown on the first die is exactly one less than the number on the second die? (L.e. what is the probability that the two numbers are consecutive?) (e) This is even trickier! What is the probability that the second die shows a number that is bigger than the one shown by the first die? Find a compact formula, in terms of n. Hint: it may help to write down the sample space as a matrix, and realizing which is the part of the matrix whose elements you are counting. Again, express the answer in terms of n