Bob and Doug play a lot of Ping-Pong, but Doug is a much better player, and wins 70% of their games. To make up for this, if Doug wins a game he will spot Bob ve 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 fteen 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 ve-point advantage Bob wins 30% of the time; he wins 30% of the time with a ten-point advantage and 30% of the time with a fteen-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. 000 P=000 000 Proportion of games won by Doug = 0