Question

(From the class on November 9) In a men's singles tennis tournament, player X and player Y are playing a match for the championship. The match is won when either player wins three out of five sets. Past statistics show that there is a 70% chance that player X will win in any one set. a) Express the match as a Markov chain, e.g. define the state of the Markov Chain and construct the transition probability matrix. b) Suggest a way to calculate, on the average, how long the match will last on the average. (Don't try to calculate the value, but just write all the necessary equations) c) If the current score is 0 set to 1 sets on player Y's favor, what is the probability that player Y will win?

Answer 1

a) The match can be expressed as a Markov chain with a transition probability matrix. b) The average duration of the match can be calculated by finding the expected number of sets played. c) The probability of player Y winning can be calculated using the current score and transition probabilities.

Step-by-step explanation:

In this problem, we need to express the match as a Markov chain and then calculate the average duration of the match.

a) Markov chain:

The states of the Markov chain will represent the scores, from 0-3, for both players. We will have a total of 16 states: (0,0), (0,1), (0,2), (0,3), (1,0), (1,1), (1,2), (1,3), (2,0), (2,1), (2,2), (2,3), (3,0), (3,1), (3,2), (3,3).

To construct the transition probability matrix, we need to determine the transition probabilities from one state to another. These probabilities can be calculated using the given information that player X has a 70% chance of winning any set. For example, the probability of transitioning from state (0,0) to (1,0) is 0.7, as there is a 70% chance that player X wins the first set.

b) Average duration of the match:

To calculate the average duration of the match, we need to find the expected number of sets played before the match is over. This can be done by multiplying the transition probabilities with the number of sets played in each state and summing them up. For example, in state (0,0), no sets have been played yet, so the expected number of sets played in this state is 0. In state (1,2), three sets have been played, so the expected number of sets played in this state is 3. We repeat this process for all states and sum up the results to find the average duration of the match.

c) Probability that player Y will win:

Given the current score of 0 sets to 1 set in player Y's favor, we need to calculate the probability that player Y will win the match. This can be done by finding the probabilities of reaching a state where player Y has won 3 out of 5 sets, starting from the current state. We can use the transition probabilities and a recursive approach to calculate these probabilities.