(13 points) There are four urns. 0 Urn 1 has 2 red balls and 8 blue balls. 0 Urn 2 has 3 red balls and 7 blue balls. 0 Urn 3 has 5 red balls and 15 blue balls. 0 Urn 4 has 12 red balls and 8 blue balls. (a) (3 points) You choose one of the urns uniformly at random and then choose a ball uniformly at random. What is the probability that you choose a red ball? (b) (3 points) You choose urn z' with probability pi, where ,; e [0, 1] for i e {1, 2, 3, 4} and 23:1 p, = 1. You then choose a ball uniformly at random from that urn. Write an expression for the probability that you choose a red ball. (c) (3 points) Again, you choose each urn i with probability p; and then choose a ball uniformly at random. Write an expression for the probability that you chose Urn 1, given that you drew a red ball. (d) (2 points) Repeat the above part assuming 1),; = i for all i E {1, 2, 3, 4}. (e) (2 points) You draw one ball from each urn independently. What is the probability that you draw only red balls