6. (20 marks) A father and his son are playing a game. Suppose that in each step of this game, the father wins with a probability 21, the son wins with a probability 41, and they end in a draw () with a probability 41. Moreover, in each step, the winner gets " +1 " point, the loser gets " -1 " point, and both get "0" point for a draw. Suppose different steps are independent. When the father gets 2 points or the son gets 1 point (in other words, the father gets -1 point), the game will end. Let Xn be the points that the father gets after the nth step. Then, {Xn} is a discrete-time Markov chain. (1) Give the state-space of {Xn} as well as the one-step transition probability matrix of {Xn}. (2) Given that the father has got 1 point, what is the probability that the game ends after exactly two steps? (3) Find the limiting probabilities of the chain. What is the probability that the father wins this game (the father wins this game if he gets 2 points)