0
$\begingroup$

In table tennis & tennis (among other sports?), a player/team must be two points ahead of their opposition to win a game. Thus, a game could technically go on for infinite deuces. However, assuming that the outcome of each rally is an independent event, with a 0.5 chance of either side winning; it is obvious that the expected number of rallies that a game will go for after a deuce is 4. This can be found (among other methods) through the equation:

a = [1/2 * 2] + [1/2 * (2+a)]

a/2 = 2

'a' is the expected number of turns that a game will go for after a deuce. A deuce has a 1/2 chance of finishing after two rallies (if both are won by either player) hence the term [1/2 * 2]. There is also a 1/2 chance of returning to deuce after two rallies, hence the term [1/2 * (2+a)].

The question arises: what would be the expected number of rallies needed to win after a deuce, if a player needed to be 3 points ahead (instead of 2)? This, too can be solved through an analogous method, yielding a result of 9.

The situation is slightly more complex if one needs to be 4 points ahead (denoted as a '4 point deuce'). This is because we must introduce a new variable, 'b', which is the expected number of rallies needed to win the deuce after one player is 2 points ahead.

We now have the equations:

a = 1/2 * (2+b) + 1/2 * (2+a) The trend seems to be that, for an 'n point deuce', the expected number of rallies needed to win it is n^2. The question is, can this be proven? b = 1/2 * (2+a) + 1/2 * 2

This yields a=16, b=12.

It seems that n^2 rallies are to be expected after an n point deuce. The question is, can this be proven?

$\endgroup$
2
$\begingroup$

Your result is correct. The situation you are describing is isomorphic to a random walk. Imagine you start at $0$ on a number line and flip a fair coin. Heads you take one step to the right (to $1$), and tails you take one step left (to $-1$). From your current position you repeat the process. In an ordinary deuce game, you stop when you get to $2$ or $-2$. In your general case, you stop when you get to $n$ or to $-n$.

It is a standard result in the theory of random walks that the expected length of this random walk is $n^2$. See for instance http://web.mit.edu/neboat/Public/6.042/randomwalks.pdf.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.