# This gives us three equations in three unknowns h1 6

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Unformatted text preview: college, 60% of freshmen become sophomores, 25% remain freshmen, and 15% drop out. 70% of sophomores graduate and transfer to a four year college, 20% remain sophomores and 10% drop out. What fraction of new students eventually graduate? We use a Markov chain with state space 1 = freshman, 2 = sophomore, G = graduate, D = dropout. The transition probability is 1 2 G D 1 0.25 0 0 0 2 0.6 0.2 0 0 G 0 0.7 1 0 D 0.15 0.1 0 1 Let h(x) be the probability that a student currently in state x eventually graduates. By considering what happens on one step h(1) = 0.25h(1) + 0.6h(2) h(2) = 0.2h(2) + 0.7 To solve we note that the second equation implies h(2) = 7/8 and then the ﬁrst that 0.6 7 h(1) = · = 0.7 0.75 8 Example 1.40. Tennis. In tennis the winner of a game is the ﬁrst player to win four points, unless the score is 4 3, in which case the game must continue until one player is ahead by two points and wins the game. Suppose that the server win the point with probability 0.6 and successive points are independent. What is the probability the server will win the game if the score is tied 3-3? if she is ahead by one point? Behind by one point? We formulate the game as a Markov...
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