Gambler's Ruin in Tennis?

Probability Level 4

Alex and Bob are playing a game of tennis. At any given time, the probability of Alex winning a point is $\dfrac{3}{5}$ , and thus Bob's chance of winning each point is $\dfrac{2}{5}$ . If $p$ is Alex's probability of winning each Game, find the value of $p$ to the nearest thousandth.

Note: In a tennis game, a player wins as soon as he/she has scored at least $4$ points AND at least $2$ more points than the opponent.. Assume it is possible for the game to continue indefinitely.

The answer is 0.736.

1 solution

Pop Wong
May 31, 2020

The possible scenario Alex wins the game:

Alex	Bob	Next Scenario	Pr	Note
40	0	Alex win next pt	Pr = $(3/5)^{3} *(2/5)$	$\frac{2025}{15625}$
40	15	Alex win next pt	Pr = $4C1 (3/5)^3 (2/5) * (3/5)$	$\frac{3240}{15625}$
40	30	Alex win next pt	Pr = $5C2 (3/5)^3 (2/5)^2 * (3/5)$	$\frac{3240}{15625}$
40	40	Alex win the deduce with Pr= $P_d$	Pr= $6C3 (3/5)^3(2/5)^3 *$ $P_d$	$\frac{4320}{15625}$ * $P_d$

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

$P_d$ = $(3/5)(3/5) + 2(3/5)(2/5)$ * $P_d$

$P_d$ = $\frac{9}{25}$ * $\frac{25}{13}$ = $\frac{9}{13}$

Pr(Alex wins) = Sum all prob. = 0.736

Gambler's Ruin in Tennis?

The answer is 0.736.

1 solution

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