Question

In a game of tennis if 2 players are tied at 40 it is called “deuce”. You must win 2 points in a row in order to win the game at this point. If you win 1 point and then lose the next then you go back to deuce. Say players A and B are at deuce and player A wins a point then they are in the state called Advantage A. If A wins the next point then the game is over and A wins. If player B wins then next point the game goes back to deuce. So we have the following states: Deuce, Advantage A, Advantage B, A wins, B wins. Suppose player A wins the serve 60% of the time. A. Create a transition matrix for this Markov chain B. If these 2 players are at deuce what is the probability that Player B wins the point?

Answer 1

Dauce o A. Transition Matrix for this Markov chain Adu. A Adv. B Game A Game B Dance 0.6 0.4 Advantago A 0,4 0.6 Advantage B 0.6 0.4 Game A o O 0 0 0 - Game B o state. state space : { Deuce, Atentage A, Advantage B, Game A, Game B The chain is Markov because the probability of moving to the next state does not depend on history prior to entering that B. The game either ends after a points or it returns to Deuce. The probability of it returning to Deuce after two points is : Pr (A wins Ist point) x Pr (Bwins and Point) + Pr/B wins Ist point) x Pr (A wins and point) = 0.680.4 + 0.480.6 0.48 Probability B wins after a points = 0.480.4 = 0.16 Probability that B wins from Deuce Probability B wing after a points + Probability B wins after 4 points + ... -0.36 + 0.16X 0.48 + 0.16X0.482 +... 0.30 77