The probability to win a Tennis match
How likely is it that the best player is going to win
Last day, I came across a post talking about how a slight percentage of probability of winning a point can get bigger in the probability of winning a match. At first, it seems like magic but thinking about the cascading multiplication of probabilities you can see that it’s logical. So, since I’m a tennis fan, I decided to dig in it and peel back the curtain on what’s behind the scene to know the exact formulas and to figure out the probability to win a Tennis match given the probability of winning a point.
Problem description
For simplification, let’s consider that in every point you have the same probability of winning and will remain fixed the entire match. We won’t consider fatigue, stamina, injuries, weather, surface type, which side on the court the player is on, serving advantages or psychological pressure due to break points, match points or audience boos.
Let p be the probability of winning any point anytime in the match. If the match consisted of just one point, your probability of winning is p. However a tennis match consist of many points, games and sets.
I suppose that you are familiar with tennis rules, here’s a small refresher:
- The scoring is as the following: 0 point called love, 1 point called “15”, 2 points called “30”, 3 points called “40” and 4 points is the game.
- To win a game, you have to win at least 4 points by a margin of 2.
- To win a set, you have to win at least 6 games by a margin of 2.
- To win a match, you have to win the majority of 3 or 5 sets (for Grand Slams).
The probability to win a Game
Let’s focus on a single game. To win it, you have to win at least 4 points by a margin of 2. There are four ways to win a game.
1- Game-Love
This one is simple, if the probability of winning a point is p, then the probability of winning 4 consecutive points is:
2- Game-15
In this case, there are 5 won points: 4 points by you and one by your opponent. There are four ways to have this situation: your opponent can win one of the first 4 points (you always win the last point). This means you won 4 points and you lost one.
3- Game-30
Just like the other games, you won 4 points, but this time your opponent won 2 points and always the last point has to come from you. So your opponent can win 2 points from 5 and we have:
4- Game-40
This one is a bit complicated. To win a game after a 40–40 (Deuce) you need to win 2 consecutive points, otherwise if you won one point (Advantage) and lost the next one you’ll start from deuce again. Let’s start by the probability of getting to Deuce. There are 6 points each one won 3 points:
Now comes the hard part, which is figuring out the probability to win deuce. The next illustration shows how to deuce situation is played.
So, to win a Deuce you need to win two consecutive points or go back to Deuce and win it. This gives
So the probability to winning a Game-40 is the probability of getting to Deuce times the probability of winning a Deuce.
Probability of winning a Game
Now that we have all the probabilities of cases to win a game, we’ll put it all together and we have:
We can see that a slight increase in probability of winning a point can drive a bigger increase to the probability of winning a game. For example if you have a probability of winning a point 70%, you’ll have a 90% chance to win to game.
The probability to win a Set
As mentioned in the problem description, to win a set, you need to win at least 6 games by a margin of 2 games. We’ll tackle this problem similarly to the game winning probability analysis. There are 7 cases to win a set.
Let g be the probability of winning a game. We have:
1- 6:0 case
This is the simplest case when you won all games in the set. Then the probability to win 6 consecutive games is:
2- 6:1 case
Similar to game probability, in this case there are 7 won games: 6 games by you and one by your opponent. Your opponent can win one of the first 6 games (you always win the last game).
The same explanation goes for the third, fourth and fifth case.
3- 6:2 case
4- 6:3 case
5- 6:4 case
6- 7:5 case
This is a special case, and the only way to win a 7:5 set is getting to 5:5 before and winning the next two games. We have:
7- 7:6 case (tie break)
This is the last case and the more complicated one, it’s a bit similar to Deuce case in the first section. The only way to win a 7:6 set is getting to 6:6 before and winning the last game which has different rules. The players are going to continue playing and they need to reach at least 7 points by margin of two otherwise they will continue to play (Oh yes, some matches can last for days!).
We will split the problem into two parts:
- Getting to 6:6
- Winning 7:6 from a 6:6
To get to 6:6 score you should pass through 5:5 then lose a game and win a one (or win a game and lose one), we have:
As mentioned above, to win 7:6 from a 6:6 score, you need to reach at least 7 points in the last game by a margin of two or continue playing until you win two consecutive points. The illustration below shows how the tie break Deuce is similar to the game Deuce.
The following are the 7 ways to win the 7:6 set: (7–0, 7–1, 7–2, 7–3, 7–4, 7–5 or winning from the tie break deuce).
We will compute the first six cases by the same rules we have computed the probability of winning games and sets because we don’t have constraints. The last case will be computed the same as the Deuce case in the game part. Let t be the probability to win a 7:6 set from a 6:6. We have:
Probability to win a set
So Now that we have computed all the sub cases probabilities we can determine the probability to win a set. Let s be this probability
We can conclude that the probability of winning a set is more sensitive to the probability of winning a point than the probability of winning a game. The curve is steeper. As an example to reach a 90% probability to win a set you just need 57% of the probability to win a point.
The probability to win a match
In tennis, there are two kinds of match: matches with the best 2 of 3 sets and matches with the best 3 of 5 sets. The former is the case for almost every tennis tournament, and the latter are the famous 4 grand slams. In this part we will tackle both since we have all ingredients prepared from the two previous sections.
Let m be the probability to win a match. For a match with best 2 of 3 sets, you can either win 2 sets — Love or 2 — 1. We have:
For a match with best 3 of 5 sets, you can win in three ways: 3 sets — Love, 3 sets — 1 or 3 sets — 2. We have:
Here is the final graph of all the probabilities.
So if Rafael Nadal have 53% probability to win a point, he’ll have 84% percent to win Roland Garros match.
The following are some examples for game, set and match winning probabilities based on p
Summary
In this blog post, I explained in detail how your odds of winning a tennis match are computed mathematically in a simple case where the probability of winning a point is the same in the match. It can get more complicated if we took into consideration all the other parameters (fatigue, stamina, injuries, serving advantages, court type, etc.)
We can conclude that the slightest advantage in odds of winning a point can become very big in winning a match. That’s why you should choose carefully your opponent.
