Ask Uncle Colin: The Most Common Tennis Score

Dear Uncle Colin,

What’s the most common score in a tennis game? Every game has either a 15-0 or a 0-15, but some games have multiple deuces.

  • Solvng Wimbledon Is A Tennis Experience, Kids

Hi, SWIATEK, and thanks for your message!

In case you’re not on top of tennis scoring, it uses a strange numbering system. Rather than counting 0-1-2-3-4 like a sensible sport, it counts love-15-30-40-game. Mathematically speaking, there are two ways a game can go: either one player scores four points before their opponent scores three, or the game reaches three points each – 40-40, also called deuce. If a game gets to deuce, it continues until one player is two points ahead of the other; the next point is “advantage”, followed by either the end of the game or another deuce.

The tl;dr is that you win by reaching at least four points with at least a two-point margin, and all of the tied scores with at least three points each are called deuce.

The question is, does the effect of having potentially multiple deuces or the effect of very often having a 15-0 dominate? It turns out, it depends on your assumptions.

For example, suppose we model tennis points as fair coin tosses. Then, the probability of hitting 15-0 is a half (obviously) and the probability of hitting 3-3 is $20\left(\frac{1}{2}\right)^3 \left(\frac{1}{2}\right)^3 = \frac{5}{16}$.

From there, it’s a 50-50 chance of reaching 4-4, and a 50-50 chance from there of reaching 5-5, and so on – so the expected number of deuces is $\frac{5}{16} \left(1 + \frac{1}{2} + \frac{1}{4} + \dots\right) = \frac{5}{8}$. It turns out that for coin flips, you expect to see deuce more often than 15-0 or 0-15.

On the other hand, if the first player wins 99% of the points, it’s pretty clear that 15-0 is the most common score.

Let’s do some algebra

You can make the algebra explicit. If the first player wins a point with probability $p$, then the game reaches 15-0 with probability $p$ and 0-15 with probability $1-p$.

How about the first deuce? That’s $20p^3(1-p)^3$ – this comes almost straight from Pascal’s triangle. Given you get there, the probability of reaching the next deuce is $2p(1-p)$, and then $4p^2(1-p^2)$ – each time, multiplying by $2p(1-p)$.

The excpected number of deuces is $20p^3(1-p)^3\left(1 + 2p(1-p) + 4p^2(1-p)^2 + 8p^3(1-p)^3 + \dots\right)$, That’s a geometric sequence! Just looking at the bracket, the first term is 1 and the common ratio is $2p(1-p)$, so its sum is $\frac{1}{1-2p(1-p)}$, ot $\frac{1}{1-2p + 2p^2}$.

That means, the expected number of deuces is $\frac{20p^3(1-p)^3}{1-2p+2p^2}$. This is not a happy fraction.

We want to know the circumstances where this is greater than both $p$ and $(1-p)$ – it’s symmetrical, so we only need to worry about $p \ge \frac{1}{2}$.

We’re not going to solve $\frac{20p^3(1-p)^3}{1-2p+2p^2} > p$ analytically. Once you cancel a $p$, it’s still a quintic which won’t permit exact solutions. However, we can apply the sledgehammer of Wolfram|Alpha to find that the two curves cross when $p \approx 0.57$.

Which means, we have three regimes:

  • If the first player wins more than 57% of their points, 15-0 is the most common score.
  • If the first player wins less than 43% of their points, 0-15 is the most common score.
  • If the rate is between 43 and 57%, then deuce is the most common score.

On a sample of two matches from yesterday, a typical value of $p$ would be about 70%. (However, the model isn’t quite as robust as it might be: the Anisimova-Noskova match had 16 deuces in 30 games, compared to eighteen 15-0s. The model would predict 9 and 21, respectively.)

An aside

One thing I’ve glossed over here: are there scores other than 15-0, 0-15 and deuce that could be more common? The answer, at least for fixed probabilities, is “no”, but that’s an exercise for the interested reader.

In conclusion

Even brushing aside the difficulties that come from real humans with their “psychology” and “resilience” and so on, it seems likely that 15-0 is the most common score in a tennis game. I’d be interested to hear whether reality reflects the maths!

Hope that helps,

- Uncle Colin