How About Some Tennis Math?
I have been thinking a lot about probability, especially the probabilistic nature of tennis. Why have I been on a probability kick lately? It certainly is a tricky and slippery subject, perhaps the hardest I have ever studied, and I guess I am just trying to keep on top of my game. Hey, you never know when a Golem or a Centaur might show up at my door with a life or death riddle based somehow on probability theory. I guess I just want to be prepared for any contingency.
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To begin, I will assume that everyone has a rudimentary knowledge of the game of tennis. I hope you all realize that the server has an advantage, especially as the players get better. As a general rule, this is more true for men than for women, particularly in the professional ranks. The server starts every point, and they know where the serve is going while the receiver has a minimal window of time to figure out the trajectory of the ball. With that in mind, let’s imagine the following scenario. A server wins 60% of their service points against a particular opponent. That means that the returner wins 40% of those points. We can ask and answer a series of questions based only on this information. Let’s get to it. For reasons that will become clear later, we will begin with a game that is already at deuce.
We will start our analysis with a player that wins 60% of all their service points against a random player. Once the game reaches deuce, the probability that the server will win the next two points is .6 x .6 = .36. That means that the server will win the next two points a little over 1/3 of the time, 36% to be exact. What about the receiver winning the next two points to break serve? That would be .4 x .4 = .16. So, 16% of the time the receiver will break serve by winning two points in a row. What is left? The only other possibility is a return to deuce. There are a couple ways we can figure this. We know that all possibilities have to add up to 1 so we can simply solve the following equation:
1- .36 – .16 = .48.
Therefore, 48% of the time the service game will return to deuce. The other way to arrive at that figure is with the following:
.6 x .4 + .4 x .6 = .48.
If you take a minute to study that short equation, you will quickly realize why it makes sense.
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Now, this gets a little tricky. With the game at deuce, what is the probability that the server will eventually win the service game? Well, they can win by winning the first two points after the first deuce, which they do at a rate of 36%. The other thing that can happen is that the game can return to deuce, which will happen 48% of the time before the server goes on to win. Therefore we end up with this equation: P = .36 +.48P. Solving for p gives an answer of .692. We now know that the server will win 69.2% of all of their service games against this opponent once the score has reached deuce.
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Notice that this player has a 60% rate of winning individual service points but ends up with a 69.2% success rate when it comes to holding their serve during a service game that has reached deuce. That is pretty interesting, maybe even a little unexpected.
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We can now break down other probabilities based on how this player does against other opponents. The first category considered is 50% of service points won. Anything less, i.e., a success rate of less than 50%, and this type of analysis is not very useful. I think we can all agree that the player’s time is better spent on improving their serve than it is doing math. For all other players that are somewhat proficient at serving, the results are as follows.
50% of points won on serve
Server will win the next 2 points 25% of the time.
Returner will win the next 2 points 25% of the time.
Game returns to deuce 50% of the time. Server wins 50% of service games that have reached deuce.
60% of points won on serve
Server will win the next 2 points 36% of the time.
Returner will win the next 2 points 16% of the time.
Game returns to deuce 48% of the time. Server wins 69% of service games that have reached deuce.
67% of points won on serve
Server will win the next 2 points 44.9% of the time.
Returner will win the next 2 points 10.9% of the time.
Game returns to deuce 44.2% of the time. Server wins 80.5% of service games that have reached deuce.
70% of points won on serve
Server will win the next 2 points 49% of the time.
Returner will win the next 2 points 9% of the time.
Game returns to deuce 42% of the time. Server wins 84.5% of service games that have reached deuce.
75% of points won on serve
Server will win the next 2 points 56.3% of the time.
Returner will win the next 2 points 6.3% of the time.
The game returns to deuce 37.4% of the time. Server wins 89.9% of service games that have reached deuce.
80% of points won on serve
Server will win the next 2 points 64% of the time.
Returner will win the next 2 points 4% of the time.
Game returns to deuce 32% of the time. Server wins 94.1% of service games that have reached deuce.
90% of points won on serve
Server will win the next 2 points 81% of the time.
Returner will win the next 2 points 1% of the time.
Game returns to deuce 18% of the time. Server wins 98.7% of service games that have reached deuce.
So, why did we start our analysis with a game that was already at deuce? That is an interesting question. We began there because the math gets a little out of hand if we start at the beginning of a player’s service game. I guess I was just trying to get everyone loose and nimble. Now that we have done our warm-up and taken off our old school polyester tops, it is time to really get down to business. Don’t worry; if the math makes your head spin, you can move on to the table. There is no crime in that.
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The probability that a tennis player holds serve (P) is equal to the probability that he holds at love (P @ love) plus the probability that they hold at 15 (P @ 15) plus the probability they hold at 30 (P @ 30) plus the probability they hold when the game goes to deuce or multiple deuces (P @ D(s)). Got it? That makes perfect sense, doesn’t it?
Now consider the following:
The probability that the player holds at love = p4
The probability that the player holds at 15 = 4p4 (1-p)
The probability that the player holds at 30 = 10p4 (1-p)
The probability that the player holds in a game that goes to deuce is much more complicated. The equation looks like this:
When the algebra is worked out, we end up with this equation:
Whew, that was a bit of work. The good news is that we can now determine how often a given player will hold serve based on the percentage of time they will win a service point. The following table contains relevant data for select service point percentages. As you can see, it also includes lots of other information. The mathematics behind the rest of the table becomes a little mind-bending, so I have decided not to add them here. You will just have to trust that the assumptions are logically based on the probability (P) that a given player will win a single service point. The rest of the table is built on that mathematical base. You will also note that I have also included the expected outcome of various matches between players serving at different rates of success.
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 80.00% | 97.80% | 52.10% | 52.80% | 54.80% | 54.20% | 55.20% |
| PLAYER B | 79.00% | 97.40% | 47.90% | 47.20% | 45.20% | 45.80% | 44.80% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 80.00% | 97.80% | 59.80% | 63.90% | 71.00% | 70.30% | 74.70% |
| PLAYER B | 75.00% | 94.90% | 40.20% | 36.10% | 29.00% | 29.70% | 25.30% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 75.00% | 94.90% | 51.90% | 52.90% | 54.00% | 54.40% | 55.50% |
| PLAYER B | 74.00% | 94.10% | 48.10% | 47.10% | 46.00% | 45.60% | 44.50% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 74.00% | 94.10% | 51.80% | 53.00% | 53.90% | 54.40% | 55.50% |
| PLAYER B | 73.00% | 93.20% | 48.20% | 47.00% | 46.10% | 45.60% | 44.50% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 70.00% | 90.10% | 51.70% | 53.10% | 53.70% | 54.70% | 55.80% |
| PLAYER B | 69.00% | 88.80% | 48.30% | 46.90% | 46.30% | 45.30% | 44.20% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 70.00% | 90.10% | 55.20% | 59.40% | 60.80% | 63.90% | 67.20% |
| PLAYER B | 67.00% | 86.10% | 44.80% | 40.60% | 39.20% | 36.10% | 32.80% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 70.00% | 90.10% | 58.50% | 65.60% | 67.50% | 72.60% | 77.30% |
| PLAYER B | 65.00% | 83.00% | 41.50% | 34.40% | 32.50% | 27.40% | 22.70% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 70.00% | 90.10% | 66.30% | 79.50% | 81.40% | 89.10% | 93.80% |
| PLAYER B | 60.00% | 73.60% | 33.70% | 20.50% | 18.60% | 10.90% | 6.20% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 67.00% | 86.10% | 53.40% | 56.50% | 57.20% | 59.70% | 62.00% |
| PLAYER B | 65.00% | 83.00% | 46.60% | 43.50% | 42.80% | 40.30% | 38.00% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 65.00% | 83.00% | 51.70% | 53.30% | 53.60% | 54.90% | 56.20% |
| PLAYER B | 64.00% | 81.30% | 48.30% | 46.70% | 46.40% | 45.10% | 43.80% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 65.00% | 83.00% | 58.20% | 66.30% | 67.40% | 73.70% | 78.50% |
| PLAYER B | 60.00% | 73.60% | 41.80% | 33.70% | 32.60% | 26.30% | 21.50% |
| serve pts | hold serve | tiebreak | tiebreak set | adv set | 3 set match | 5 set match | |
| PLAYER A | 62.00% | 77.60% | 51.60% | 53.40% | 53.60% | 55.10% | 56.30% |
| PLAYER B | 61.00% | 75.60% | 48.40% | 46.60% | 46.40% | 44.90% | 43.70% |
Take some time to study this table. You will find that a 1% advantage in service points won translates into a much bigger chance of winning sets and matches. The table is quite interesting and informative. I will be posting many more essays on this topic in the future. Stay tuned…
