Question 6 [25 pt]: You are playing a simplified game of Wheel of Fortune. The objective is to correctly guess a three letter word. Let X, Y, and Z represent the first, second, and third letters of the word, respectively. There are only 8 possible words: X can take on the values c' or'. Y, can be 'a' or o', and Z can be b' or t'. Before you guess the word, two of the three letters w be revealed to you. In the first round of the game, you choose one of X Y or Z to be revealed. In the second round, you choose one of the remaining two letters to be revealed. In the third round, you guess the word. If you guess correctly, you win. The utility of winning is 1, while the utility of losing is You watch the game a lot and determine that the eight possible words occur with the probabilities shown on the right. Your goal is to act in such a way as to maximize your chances of winning (and thereby your expected utiity) 0.10 0.10 co 0.20 0.20 l a0.05 0.15 l 0.05 0.15 1. What is the distribution P(X, Y)? Your answer should be in the form of a table 2. Are the first and second letters (X and Y) independent? Show a specific computation that supports your claim 3. Are the second and third letters (Y and Z) independent if you know the value of the first letter (X)? Show a specific computation that supports your claim 4. Suppose that in the first round, you ask about X and are told that X- c. It is the second round and you can now either ask the host to reveal Y or to reveal Z. If you ask the host to reveal Y, what is the probability that you will win in the third round? What letter should you ask the host about in the second round to maximize your chance of winning, Y or Z