Clash of the Tennis Titans

Probability Level 3

Roger Federer and Rafael Nadal are playing a tennis match.

It is Roger's serve.

Because he is serving (and the server usually has an edge over the receiver), 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.

The answer is 0.855967.

1 solution

Pop Wong
May 17, 2020

The possible scenario Rogor won the game:

Federer	Nadal	Next Scenario	Pr	Note
40	0	Federer win next pt	Pr = $(2/3)^{3} *(2/3)$	$\frac{144}{729}$
40	15	Federer win next pt	Pr = $4C1 (2/3)^3 (1/3) * (2/3)$	$\frac{192}{729}$
40	30	Federer win next pt	Pr = $5C2 (2/3)^3 (1/3)^2 * (2/3)$	$\frac{160}{729}$
40	40	Federer win the deduce with Pr= $P_d$	Pr= $6C3 (2/3)^3(1/3)^3 *$ $P_d$	$\frac{160}{729}$ * $P_d$

$P_d$ = Pr(Roger wins next two pt) + Pr(Re-deduce) * $P_d$

$P_d$ = $(2/3)(2/3) + 2(2/3)(1/3)$ * $P_d$

$P_d$ = $\frac{4}{9}$ * $\frac{9}{5}$ = $\frac{4}{5}$ ; you may calculate $P_d$ by G.P.

Pr(Roger win) = $\frac{144}{729}$ + $\frac{192}{729}$ + $\frac{160}{729}$ + $\frac{160}{729}$ * $\frac{4}{5}$ = $\frac{144+192+160+128}{729}$ = $\frac{624}{729}$ = 0.855967

Clash of the Tennis Titans

The answer is 0.855967.

1 solution

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