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I roll a fair die until I get my first ace. Let X be the number...

I roll a fair die until I get my first ace. Let X be the number of rolls I need.

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)

(c) Find P(X = Y).

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

HINT: What are 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.

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