# Random Walk Probability - Tennis Match

You and an opponent are playing tennis - first to get $$2$$ wins in a row wins. The probability of you getting a win is $$0.6$$. The probability of him getting a win is $$0.4$$. What's the probability of you winning the game?

I think this can be modeled as a Markov chain with 5 states (2 Losses, 1 Loss, 0 net, 1 Win, 2 Wins). Therefore, I think I could write out some equations to solve this. Can someone tell me if this makes sense/if it's wrong?

P(you win right off the bat) $$= (0.6)(0.6) = 0.36$$

P(he wins right off the bat)$$= (0.4)(0.4) = 0.16$$

P(you win)$$= \frac{0.36}{0.36+0.16}$$

• You are right in your answer and look at other way of solving as there are numerous ways of solving it including markov chain but those solutions are pretty sophisticated for this problem Aug 15, 2020 at 17:01

Case 1: You win two games consecutively$$= 0.36$$

Case 2: You win a game and your opponent loses a game$$= 0.24$$

Case 3: You lsoe a game and your opponent wins a game$$= 0.24$$

Case 4: You lose two consecutive games and your opponent wins $$= 0.16$$

In both cases 2 and 3, the game can be viewed as draw and back to square one. Thus the probability that is not a winner is sum of case 2 and 3 $$= 0.48$$

The probability that you will win $$= 0.36 + 0.48*(.36)+0.48^2*(.36) + \cdots \infty$$

$$= 0.36\frac{1}{(1-0.48)} = \frac{9}{13}$$

The probability that your opponent will win $$=0.16 + 0.48*(.16)+0.48^2*(.16) + \cdots \infty$$

$$= 0.16\frac{1}{(1-.48)} = \frac{4}{13}$$

This is one way you can simplify the game and find the solution unless you know the Markov Chain way of solving.

• Thank you so much! Aug 15, 2020 at 17:09
• You are welcome Aug 15, 2020 at 17:13
The text passage contains a mathematical question about the probability of winning a tennis match under specific conditions, and it provides a detailed answer using different cases and calculations.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Random Walk Probability - Tennis Match

You and an opponent are playing tennis - first to get $$2$$ wins in a row wins. The probability of you getting a win is $$0.6$$. The probability of him getting a win is $$0.4$$. What's the probability of you winning the game?

I think this can be modeled as a Markov chain with 5 states (2 Losses, 1 Loss, 0 net, 1 Win, 2 Wins). Therefore, I think I could write out some equations to solve this. Can someone tell me if this makes sense/if it's wrong?

P(you win right off the bat) $$= (0.6)(0.6) = 0.36$$

P(he wins right off the bat)$$= (0.4)(0.4) = 0.16$$

P(you win)$$= \frac{0.36}{0.36+0.16}$$

• You are right in your answer and look at other way of solving as there are numerous ways of solving it including markov chain but those solutions are pretty sophisticated for this problem Aug 15, 2020 at 17:01

Case 1: You win two games consecutively$$= 0.36$$

Case 2: You win a game and your opponent loses a game$$= 0.24$$

Case 3: You lsoe a game and your opponent wins a game$$= 0.24$$

Case 4: You lose two consecutive games and your opponent wins $$= 0.16$$

In both cases 2 and 3, the game can be viewed as draw and back to square one. Thus the probability that is not a winner is sum of case 2 and 3 $$= 0.48$$

The probability that you will win $$= 0.36 + 0.48*(.36)+0.48^2*(.36) + \cdots \infty$$

$$= 0.36\frac{1}{(1-0.48)} = \frac{9}{13}$$

The probability that your opponent will win $$=0.16 + 0.48*(.16)+0.48^2*(.16) + \cdots \infty$$

$$= 0.16\frac{1}{(1-.48)} = \frac{4}{13}$$

This is one way you can simplify the game and find the solution unless you know the Markov Chain way of solving.

• Thank you so much! Aug 15, 2020 at 17:09
• You are welcome Aug 15, 2020 at 17:13
user
The text passage contains a mathematical question about the probability of winning a tennis match under specific conditions, and it provides a detailed answer using different cases and calculations.

TYPE: SINGLE_QUESTION_WITH_ANSWER