Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 80% 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 fifteen points, and continue to spot him fifteen points as long 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 five-point advantage Bob wins 20% of the time; he wins 50% of the time with a ten-point advantage and 60% of the time with a fifteen-point advantage.
Model this situation as a Markov chain using the number of consecutive games won by Doug as the states. There should be four states representing zero, one, two, and three 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.
find matrix p and vector s and the proportion of games won by doug
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