Suppose that whenever you have an odd amount (2k +1) of money, you bet (k+1) on a dodgy coin toss (0.52 probability of tails, 0.48 probability of heads). Each time, you double your bet as profit with a heads, and lose your bet with tails. Suppose that whenever you have an even amount (2k) of money, you bet k on a similar coin toss. You start with 3. As soon as you have 0 money left, you stop. Also, if you have more than 6 money, you stop gambling. (a) Model the problem as a directed graph with probability-edge- weights. Make sure that the arcs exiting each state have weights summing to 1. (b) Give the transition matrix for the Markov chain defined in (a). (c) Using (b), give the probability of having 2 after two coin tosses. (d) List the communication classes of the Markov chain, and for each class, determine (with proof) whether the communication class is recurrent or transient. (e) Does the Markov chain have a time-reversible distribution? Prove or disprove. (f) Does the Markov chain have a unique stationary distribution? Prove or disprove. (g) Calculate the probability of the game ending with you having more than 6.