Problem 3. Consider a game between 3 friends (labeled as A, B, C). The players take turns (i.e., A→ B→C → A→B→C...) to flip a coin, which has probability p = (0, 1) to show head. If the outcome is tail, the player has to place 1 bitcoin to the pool (which initially has zero bitcoin). The game stops when someone tosses a head. He/she, which is the winner of this game, will then earn all the bitcoin in the pool. (a) Who (A, B, C) has the highest chance to win the game? What is the winning prob- ability? Does the answer depend on p? What happens if p → 0? (b) Let Y be the amount of bitcoins in the pool in the last round (of which the winner will earn all). Find E[Y] and Var(Y). (c) Let Z be the net gain of Player A of this game (that is, the difference of the bitcoins he earns in this game (0 if he doesn't win), and the total bitcoins he placed in the previous rounds). Find E[Z]. (d) † Repeat (b), (c) if the rule of placing bets is replaced by "the player has to place k bitcoins to the pool at k-th round