Gambling Woes Forest proposes a gambling game to you (uh oh!). Every day, you flip two independent fair coins. If both of the coins come up heads, then your fortune triples on that day. If one coin comes up heads and the other coin comes up tails, then your fortune is cut in half. If both of the coins comes up tails, then game over: you lose all of your money! Forest claims that you can get rich quickly with this scheme, but you decide to calculate some probabilities first. (a) Let M0 denote your money at the start of the game, and let Mn denote the amount of money you have at the end of the nth day. Compute E[Mn+1 | Mn]. (b) Use the law of iterated expectation to calculate E[Mn+1] in terms of E[Mn]. Solve your recurrence to obtain an expression for E[Mn+1]. Do you think this is a fair game? (c) Calculate P(Mn > 0). What is the behavior as n → [infinity]? Would you still play this game?