Two players are playing a sequence of points, which begin when one of the players serves. Suppose that player 1 wins each point she serves win probability $p$ and wins each point her opponent serves with probability $q$. Suppose the winner of a point becomes the server of the next point.
(a) Find the proportion of points that are won by player 1.
(b) Find the proportion of time that player 1 is the server.
I am stuck in how to find the transition probability. I feel like we need to define two Markov chains for the question (a) and (b). For (a) I feel like to define the $X_n$ is the $nth$ serves player 1 won. And $P_{1,1}=1-p$ $P_{1,2}=1-p$. Does it work? Thanks.
Best Answer
HINT
I interpret the question to mean the players play forever, and "proportion" means the long-term average of the fraction (of points in (a), of time in (b)).
Answer (a) = answer (b). Can you see why?
Given the above, you really only need to answer one question. I find it more natural to think of a $2$-state M.C. where state $i$ represents player $i$ serving. What are the $4$ transition probabilities $P_{11}, P_{12}, P_{21}, P_{22}$?
Once you set up the M.C. the long-term average is just the steady state distribution. Can you find that?
Can you finish from here?