Problem 4 (25 points): Urns with balls Consider an urn containing 6 red balls and 4 blue balls. Two players A and B play a game as follows: in round 1, player A picks 3 balls at random from the urn, notes the colors, and return them to the urn. In round 2, player B picks 3 balls at random, notes the colors, and then return them back to the urn, and so on. A's objective is to obtain 3 red balls, and B's objective is to obtain 2 red balls and 1 blue ball. If a player reaches his or her objective at one round, the game ends (therefore, the game can end in the first, second, third, etc. round). (i) (6 points) Find the probability that A picks a winning set of balls at a round. Find the probability that B picks a winning set of balls at a round. (ii) (7 points) Find the probability that A wins the game, that is, he is the first to win a round. Hint: for this and the next question you may want to derive a recursive formula. (iii) (6 points) Find the expected number of rounds played until one player wins. (iv) (6 points) Assume that the game changes as follows: at each round, player A picks 1 ball and notes the color, then, without returning the ball back to the urn, player B picks a ball and notes the color, then without returning the ball, player A picks another ball, and so on until both players have picked 3 balls. Assume the players have the same objective as before. Similar to (i) above, find the probability that A and the probability that B pick a winning set of balls at a round. Hint: We are only concerned about a single round. Don't be shy to dive into combinatorics for this one. Solution (You can use the back of the page too.)