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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.

1. 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.