Model the following problem using Markov Chains: A, B, and C roll a pair of dice in turn. A wins if she gets a sum of 11, B wins if she gets a sum of 10, and C wins if she obtains a sum of 9. The game ends when a player wins. (25 pts) (a) write down the transition probability matrix if the players roll with the sequence A, B, and C. Is the Markov Chain ergodic? If not what are the absorbing states, as well as the Q and R matrices? (b) find the winning probabilities for all players if the game starts with A (c) Now consider the following modification: each time the player who will roll next is determined by the roll of a fair dice (from the beginning). If the dice is 3 or less player A rolls next, if it is 4 or 5, B rolls next, and C rolls if the dice shows 6. For this version write down the transition probability matrix. (d) In part (c), if you have the chance to choose one of the positions, which one you will choose and why? (make necessary computations)