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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}$

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  • $\begingroup$ 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 $\endgroup$ Aug 15, 2020 at 17:01

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Answer:

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.

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  • $\begingroup$ Thank you so much! $\endgroup$ Aug 15, 2020 at 17:09
  • $\begingroup$ You are welcome $\endgroup$ Aug 15, 2020 at 17:13

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