The question reads: You roll a fair die until you obtain a 6. Your opponent then rolls the same die until he or she rolls an even number. Find the probability that you roll the die more times than your opponent.
My attempt: Let X be the number of rolls until a six is shown, and let Y be the number of rolls until an even number is shown. Then, X and Y both follow geometric distributions with p = 1/6 and p = 1/2, respectively. I found the expected value (6 and 2, respectively), but now I'm stuck. Any help on where to go from here or whether I should just start over?