You and your friend play a betting game where you start with 1 dollar and your friend starts with N dollars, where N is a natural number. Each round, you 'flip a fair coin for the shortest current stack' (e.g. you win the 'shortest stack amount' from your friend if it lands Heads and your friend wins the 'shortest stack amount' from you if it lands Tails). You buy back in for an extra $1 every time you lose your current stack to your friend and the game continues, but if your friend loses all of his stack to you, he doesn't buy back in and the game ends. (a) What is the expected number of rounds that this game will last? (b) What is the expected amount of profit that you walk away with? (c) What's the expected amount of times you expect to buy back in for an iteration of the game for very large N? (d) In the real world, a U.S. penny has a ~51% chance of landing the same side-up it was before it was flipped up into the air, but an ~80% chance of landing Tails if it's spun from its edge on a tabletop. So, let's change the game from now on so you can also use your real-life U.S. penny instead (if it somehow lands on its edge, just flip/spin it again): you either flip the coin with the Heads side initially facing up (51% Heads), or with the Tails side initially facing up (49% Heads), or you can spin it on a tabletop (20% Heads). Or, you can stick with your perfectly fair 50% Heads coin whenever too. Your objective is always to maximize your expected profit. What is your optimal strategy, and what is the expected amount of profit that you walk away with? (e) The game also ends when have lost N dollars yourself (you lose when you're down N dollars from your original 1 dollar buy-in), in which case your friend wins. What is the minimum probability of landing Heads that the coin must have for you and your friend to have equal chances of winning the game?