You and your friend play a betting game where you start with $1 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' (i.e., 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 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 is the expected number 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 about a 51% chance of landing the same side up as before it was flipped, and about an 80% chance of landing Tails if spun on edge. Now, you may choose to use your real U.S. penny in the game: flip it with Heads up (51% Heads), Tails up (49% Heads), or spin it (20% Heads). Alternatively, you can use the perfectly fair coin (50% Heads). Your goal is always to maximize your expected profit. What is your optimal strategy and the expected profit? (e) The game also ends if you lose N dollars (i.e., you are down N dollars from your original $1 buy-in), in which case your friend wins. What is the minimum probability of landing Heads the coin must have for you and your friend to have equal chances of winning the game?