# Clash of the Tennis Titans

Discrete Mathematics Level pending

Roger Federer and Rafael Nadal are playing a tennis match.

It is Roger's serve.

Because he is serving (and the server have an edge over the reciever), Roger's odds of winning a point are $$2:1$$ in his favour.

Find the probability that Roger wins his service game. Round your answer off to 3 decimal places.

$$\textbf{Details about Tennis Scoring}$$

• For those who are not acquainted with tennis, the scoring is explained on this page. Briefly, the game scoring (the scoring that concerns us in this problem) is as follows

• $$0$$ is also called "love" in tennis.

• Examples of scores - $$0-0, \ 0-30, \ 40-15$$ etc.

• $$40-40$$ is called Deuce.

• Once at deuce, a player must win $$\textbf{2 consecutive}$$ points to take the game.

• If Player $$A$$ $$\textbf{wins the deuce point}$$, then the score is $$\textbf{Advantage Player A}$$.

• If, after gaining advantage, the player $$\textbf{looses the next point}$$, then the score comes $$\textbf{back to deuce}$$.

• In all the points that concern us, Roger serves as they are all part of the same Federer service game.

• This is a completely hypothetical scenario.

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