Suppose that we roll a pair of ordinary six-sided dice repeatedly until either a roll of either 4 or a 7 occurs. (If another outcome occurs, the dice are rerolled, as often as necessary.) What is the probability that the final roll is a 4?

Now suppose that you pay a casino $1 to play this game. If the final roll is a 4, you receive $X. If the casino wants to operate this game without a "rake" (so that it neither gains nor loses money on average), what should X be?