A boy and a girl move into a two-bar town on the same day. Each night...

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A boy and a girl move into a two-bar town on the same day. Each night the boy visits one or another of the two bars, starting in bar 1 according to a Markov chain with transition matrix (10) 0.7 0.3 0.3 0.7 Likewise, a girl visits one or the other bar with transition matrix 0.4 0.6 0.6 0.4 :). starting in bar 2. Assume that the two Markov chains are independent. Naturally the game ends when boy meets girl, i.e., when they go to the same bar. a) Argue that the progress o fthe game can be described by a three-state Markov chain where one state is abso representing the end of the game ,and the other two states give the different bar identities of the boy and the girl. Exhibit the transition matrix for this chain. (4) b) Let N denote the number of the night on which boy meets a girl. What is the distribution of N? (6) A boy and a girl move into a two-bar town on the same day. Each night the boy visits one or another of the two bars, starting in bar 1 according to a Markov chain with transition matrix (10) 0.7 0.3 0.3 0.7 Likewise, a girl visits one or the other bar with transition matrix 0.4 0.6 0.6 0.4 :). starting in bar 2. Assume that the two Markov chains are independent. Naturally the game ends when boy meets girl, i.e., when they go to the same bar. a) Argue that the progress o fthe game can be described by a three-state Markov chain where one state is abso representing the end of the game ,and the other two states give the different bar identities of the boy and the girl. Exhibit the transition matrix for this chain. (4) b) Let N denote the number of the night on which boy meets a girl. What is the distribution of N? (6)

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