You roll a fair six-sided die repeatedly. After each roll you immediately receive an amount equal to the face value. If you roll 4, 5, or 6 you must roll again; the game stops the first time you roll 1, 2, or 3. Derive the expected total payout. Show all steps (conditioning on the stopping time or using an infinite series). Then generalize to a biased die where P(roll in {4,5,6}) = p and the expected value of a single roll is μ; express the expected total payout in terms of p and μ. Finally, suppose each roll after the first incurs a fixed cost c; derive the break-even c that makes the game fair.