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 rst, second, third, etc. round). (i) 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) Find the probability that A wins the game, that is, he is the rst to win a round. Hint: for this and the next question you may want to derive a recursive formula. (iii) Find the expected number of rounds played until one player wins. (iv) 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, nd the probability that A and the probability that 3 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