• # The Mathematical Reason for Deuce

• ## and other intriguing insights into the world of tennis scores

Last  Monday morning at Mathspace HQ we started the day with an impassioned  debate. It was about the first Grand Slam for 2018 — the Australian  Open.

What  initially started as a chat about the Kyrgios/Dimitrov duel that had  unfolded the night prior, ended in a more general debate about the  origins of the unique (and, let’s be honest, weird) tennis scoring  system.

Love, 15, 30, 40, deuce, advantage….

What does it all mean? And can we think about any of this mathematically?

#### It all starts with love (or l’oeuf)

In tennis, if you’ve scored no points, your score isn’t zero. It’s love.

‘Love’ originally comes from the French word ‘l’oeuf,’ meaning egg. An egg looks like a zero.

Somehow, over time, English speakers turned “l’oeuf” into “love”. And it stuck.

#### 15, 30, 40…

If you  win a point your score isn’t ‘one’. Instead, your score goes up to 15,  and then 30. But after 30 it goes to 40, instead of 45 (as you’d  expect).

Now the jury is out on the reason for this one, but there are a few possibilities.

My  personal favorite is that early in the game’s history you only needed  four points to win a game, and scores were marked on a clock. So you  would start at 0 for the beginning of an hour and then go up 15 minutes,  30 minutes, 45 minutes and 60 minutes which is the end of the hour.

But these aren’t quite the scores today. So what happened?

It’s  possible that 45 was changed to 40 just because it’s easier to say in  French. But that would be too boring. I prefer to think that it was  changed to 40 to make space for a new score when the deuce rule was  introduced.

Deuce was introduced to reduce the impact of the lucky shot…

#### Deuce

If both  players get to 40, the score becomes deuce. At this point, if you win a  point you get to advantage, and you need to win another point to win  the game. If you lose a point when you are on advantage, the score goes  back to deuce. Deuce effectively means you need 2 points in a row to win  the game. So I’d say that Advantage is 50 and Game is 60 on our clock  point scoring system.

#### But why introduce ‘deuce’ in the first place? There is a mathematical answer!

The short answer is that deuce makes tennis fairer.

Let’s  assume that each player has a constant chance of winning any point.  Suppose we have two players, Amis and Vauquelin. Amis is a stronger  player, and so has a 70% chance of winning each point. Vauquelin  consequently has a 30% chance to win a point.

Without  the deuce rule, if both players get to 40, Amis has a 70% chance to win  the game on the next point and Vauquelin has a 30% chance.

With  the deuce rule, if both players get to 40, then they would need to win  two consecutive points to win the game, since if they each won one point  they would go back to deuce. So the probability of winning the game at  this point is the probability of winning one point, squared. For Amis  this becomes 49% and for Vauquelin this becomes 9%.

The  remaining 42% is the probability that the score returns to deuce. Since  each deuce is effectively the same situation, we can find the  probabilities of winning after any number of deuces by dividing each of  the probabilities above by the probability of either player winning.  This gives Amis approximately 84% chance to win, and Vauquelin  approximately 16%.

This  is quite a dramatic change from 70% and 30% respectively. The stronger  player has an even greater chance to win than their initial probability  would indicate making for a fairer match by reducing the element of luck  and rewarding the actual stronger player. And this reason extends to  all the scoring rules in tennis.

While  the calculations are a little complicated for this blog post, we can  even work out the probabilities of winning from the start of a game.  Without the deuce rule, Amis has a probability of around 87% and  Vauquelin has a probability of around 13%. With the deuce rule, Amis has  a probability of around 90% and Vauquelin has a probability of around  10%.

It’s  worth noting that all of these numbers are based off a fairly  simplistic assumption. The probabilities are more likely to change with  each point. For example, some players are stronger when serving and  others are stronger when receiving. That said, there’s an entire branch  of mathematics called stochastics which answers questions like this.

#### Fun fact

The longest single game of tennis had 37 deuces! Assuming each player had a 50:50 chance of winning each point this was approximately a 1 in (1.9 x 10²³) chance of happening.

Thanks to our resident mathematician (and now, tennis expert), David Groenhout, for this post!

Did  you enjoy this post? Check out another piece written by our resident  surfer, Jake Horder, on calculating the distance of the horizon!