You roll a fair die until you get your first ace. Let Y be the number of rolls you need.

(a) Find P( X+Y = 8)

HINT: Suppose you and I roll the same die, with me going first. In how many ways can it happen that X+Y = 8, and what is the probability of each of those ways?

(b) Find P(X+Y >= 8)

HINT: See the previous HINT.

(c) Find P(X = Y).

HINT: CRAPS PRINCIPLE. Or sum up the probabilities of the ways this can happen.

(d). Find P(X > Y).

HINT: One way is to use obvious properties of the three numbers P( X > Y), P(X < Y), and P(X=Y).

Another way is to sum over the possible values of Y.

(d) Find P(X > Y+2).

HINT: One way is to use the values of P(X >Y) and P(X > Y+2 | X > Y).Or, for a more grind-it-out solution, sum the probabilities of all (x,y) values for which the event X > Y+2 happens.