What has 31⁄88 to do with tennis?
– an observation that is nearly trivial, but not quite
In a best‐of‐5‐sets tennis match that provides for best‐7‐of‐12‐points tiebreaks in all sets, you can win the match while winning only 31⁄88 of the points. You cannot win a tennis match played under any standard set of rules while winning a smaller proportion of the points.
Specifically, you can win the match 0–6, 0–6, 7–6, 7–6, 7–6. The first four sets can of course take place in any order. The games you win, you win to 30. The games you lose, you lose to 0. The tiebreaks, you win 7–5. You win 93 points out of 264, a fraction of 31⁄88. I omit the arithmetic.
That would be utterly boring were it not for a curious twist in the arithmetic. In what follows, the word minimal means “done in such a way as to win as small a proportion of the points as possible”. The words you and your refer to the winner of the match.
I asserted that the 3 sets that you win, you win 7–6,
rather than 6–4. We cannot reach that conclusion on
any kind of principle: we have to do the arithmetic. The interesting twist is that if
there were such a thing as a best‐of‐7‐sets match, then your
minimal win would be 0–6, 0–6,
0–6, 6–4,
6–4, 6–4,
6–4, with scores of 6–4, rather than
7–6, in the sets you win. The reason is as
follows. In a best‐of‐5‐sets match with
no tiebreaks, your minimal win is by 0–6,
0–6, 6–4,
6–4, 6–4, and you win
72 points out of 204, a fraction of about
0.353. In a best‐of‐7‐sets match
with no tiebreaks, your minimal win would be by 0–6,
0–6, 0–6,
6–4, 6–4,
6–4, 6–4, and you would win
96 points out of 280, a fraction of
about 0.343. In either case, 5
sets or 7 sets, if you replace a set minimally won
6–4 by a set minimally won 7–6, then you
introduce 20 more points, of which you win 7.
That changes the proportion of points you win, by pulling it slightly towards
7⁄20 or 0.350. That is, downwards
from 0.353 in