Can mathematics help you to win the US Open?

We are almost half way the last Grand Slam of the year, the US Open. Top players like Federer, Nadal and Djokovic spend hours per day on court to raise their game to the highest possible level. The ultimate goal is to win the title. By serving smarter, players can considerably increase their chances of winning.

Serving smart

Tennis is the only sport in which players are allowed to serve twice. In top tennis, serving is a big advantage, especially for men. Players take more risk on their first serve, and win most of the points on average. If the first serve is not correct, a more conservative second serve follows. Also after a second serve, on average, top players win more than half of the rallies. Frank Klaassen and Jan Magnus have calculated that both for men and women, about 60% of the first serves are correct, and 86% of the second serves. If the serve is good, men win on average 74% of the rallies after a first serve, and 59% of the rallies after a second serve. For women, these percentages are 63% and 53%. Players take significantly less risk at the second serve. This makes sense, because when they hit a double fault, they lose the point right away. Or not? The probability to win the rally after a second serve is a lot smaller than after a first serve. Is it wise to play safe and avoid a double fault? Or is it smarter to hit some more double faults, but to win more points in the rally? The finals of Wimbledon and the Australian Open of this year show there is a lot to gain. Andy Murray could have doubled his win chances against Djokovic by serving smarter.

The Wimbledon final 2015

In the Wimbledon final of this year, Novak Djokovic and Roger Federer faced each other. A final many people had hoped for, containing the numbers one and two seeded. Novak Djokovic won by 7-6 6-7 6-4 6-3. Could Federer have beaten Djokovic by serving smarter? Let’s first take a look at the statistics:

Tabel1 engels

Both men played approximately two out of three first serves correctly. If the first serve was correct, they won three out of four points. The biggest difference was seen on the second serve. Both played a conservative second serve. Djokovic hit only one double fault in the entire match, Federer three. But Federer won significantly less points on the second serve compared to Djokovic, 52% versus 61%. Let us assume that the players can vary the amount of risk they take. And they can hit a serve somewhere in between the first and the second serve. The probability that the serve is correct, and the probability to win the rally if the serve is correct, lie between the first and second serve characteristics. It is possible to calculate how much risk one should take on the first serve and the second serve, to maximize the probability to win the point. Graham and Geoff Pollard have investigated how to do this, and the great thing is that you only need high school mathematics to do so. Do you want to know how? Click here. Roger Federer his first serve was optimal, but he could have better taken more risk on his second serve. It would have led to more double faults. But he would have won more points in total, because his chances in the rally would have increased. Novak Djokovic his serving tactic was nearly optimal.

Figuur 1 engels

Suppose that Roger Federer had served tactically optimal. What impact would it have had on his chance of winning? Novak Djokovic won 69.2% of the points when he served. Roger Federer won 65.8%. By taking more risk on the second serve he could have raised that number to 66.6%. Suppose Djokovic and Federer face each other again at the US Open. And assume they both play equally well as last time, only Federer now takes more risks on his second serve. How will the chances of winning a game, a set or the match change? Djokovic would still have the best odds, but Federer would increase his chance of winning by more than 4%! In a thrilling match, serving smart can make the difference between winning and losing. 

Tabel 2 engels

Figuur 2

Federer congratulates Djokovic with his victory. Will it be the other way around next time?

The Australian Open final 2015

In the final of the Australian Open, Novak Djokovic and Andy Murray faced each other. Djokovic also won this match, by 7-6 6-7 6-3 and 6-0. For Djokovic, something strange happened in this match. He won more points on his second serve than on his first serve. Andy Murray had, like Roger Federer in the Wimbledon final, a big difference between his first and second serve. On his second serve, he won only 37% of the rallies. For him it would have been wiser to take a lot more risk on his second serve.

Figuur 3 engels

If he had taken more risk on the second serve, the probability to win a point would have increased from 54.2% to 57.1%. If Djokovic would had served the same, the probability for Murray to win the match would have increased from 16.6% to 31.2%. Almost a factor two!

Tabel 3 engels

The US Open

The finals of Wimbledon and the Australian Open showed big differences between players. For Novak Djokovic, there was relatively little difference between his first and second serve. For Roger Federer and Andy Murray, tactically, it would have been better to take more risks on the second serve. This may be true in general, but the optimal strategy can also depend on how well one plays on a given day, and on the opponent. At the US Open, the players should therefore keep track of how many points they score on the first or second serve. And adapt their strategy during the match. For a player, this is difficult because they have to concentrate on their play. The coach could do this, but unfortunately he is not allowed to give tips. Although this happens in practice, because the coach is in the stadium. A white cap means more risk, a black one means less risk?

Sources

Franc Klaassen and Jan Magnus: Analyzing Wimbledon, The power of statistics

Graham Pollard and Geoff Pollard: Optimal Risk taking on first and second serves

Geoff Pollard: What is the best serving strategy?

Paul Newton and Joseph Keller: Probability of winning at tennis. I. Theory and data

Statistics from www.flashscore.nl

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