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#### Question 9

Bob and Doug play a lot of Ping-Pong, but Doug is a much better player and wins 60% of their games.

To make up for this, if Doug wins a game he will spot Bob five points in their next game. If Doug wins again he will spot Bob ten points the next game and if he still wins the next game he will spot him 15 points, and continue to spot him 15 points as he keeps winning.

Whenever Bob wins a game he goes back to playing the next game with no advantage. It turns out that with a 5 point advantage Bob wins 60% of the time; He wins 90% of the time with a 10 point advantage and 90% of the time with a 15 point advantage.

Model this situation as a Markov chain using the number of consecutive games won by Doug as the states.

There should be 4 states representing 0, 1, 2 and 3 or more consecutive games won by Doug.

Find the transition matrix of this system, the steady state vector for the system, and determine the proportion of games that Doug will win in the long run under these conditions.

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