We're likely to see penalty shootouts at the Euro 2012. Most shooters have a 75 percent chance of scoring. But according to binomial probability, and common preconceptions, the Germans are even better.
For a world class soccer player, scoring a goal from a short distance of just 11 meters is easy. Just aim for the upper right or left hand corner of the goal and kick the ball at 100 kilometers (60 miles) per hour.
It should take less than half a second, or precisely 0.4 seconds, for the ball to reach the goal line. So, the goalkeeper has 0.2 seconds to see where the ball is coming from and another 0.2 seconds to react.
The ball travels so fast that no one really expects the keeper to save the goal - the odds of scoring heavily favor the shooter. And it is not only down to the speed. The goal is about four times bigger than the average goalie. It is almost impossible to cover the entire goal.
Even the best miss
And yet some of the world's best soccer players do miss penalties. Maradona, Ronaldo, David Beckham, Michel Platini, Roberto Baggio and Uli Hoeneß have all failed to score from 11 meters.
"It's sometimes unbelievable that they don't hit it, you can imagine the pressure they have to take. If they really could do it say like a robot, a robot would have a 100 percent chance, but a penalty taker is not a perfect robot" said Metin Tolan, an experimental physicist at Dortmund's Technical University, and author of a book on the mathematics in soccer.
Although it is hard to recreate the immense psychological pressure of a real match during training, it is possible for players to practice technique.
The precise placement of a shot is critical, since a tiny error can trigger a very big miss.
"If you change the initial speed of the ball by only one kilometer an hour, and the initial launching angle of the ball by one degree, you have a spread in height of 50 centimeters, so the risk to shoot over the goal is enhanced," Tolan said in an interview with DW.
A numbers game
The entire surface area of a goal is 2.44 x 7.32 meters, or about 18 square meters.
If the height of the goalie with his arms outstretched is 2.25 x 2 meters, the goalie can cover an area of 4.5 meters.
That is one-quarter of the goal's surface.
"Seventy-five percent of the goal cannot be covered by the goalie," said Tolan. "This can explain why you have a 75 percent goal probability for a penalty from 11 meters."
In 1976, the penalty shootout was introduced at the European Championships to decided matches that end on a draw after the regular 90 minutes of playing time and thirty minutes of overtime.
Since then, the analytical website sportingintelligence.com says 343 penalties have been taken and 258 goals have been scored.
This is an overall success rate of 75 percent.
But 75 percent is an average, so some national teams do better than others, notably the Czechs and Germans. Others do worse, especially the Netherlands and England.
At the 1976 European Championship final between Czechoslovakia and West Germany - the first ever penalty shootout in an international tournament - striker Uli Hoeneß's shot flew over the crossbar. Then, Czech midfielder Antonin Panenka tricked German goalie Sepp Maier into diving to the corner and chipped the decisive kick down the middle. It was the first and only time that the Federal Republic of Germany had lost in a penalty shootout.
Average success versus German success
To date, Germany has scored a total of 26 out of 28 penalties. So, it has missed only two penalties, giving it a success rate of 93 percent.
The second and last time Germany went down a penalty was 30 years ago.
It was at the 1982 World Cup semi-final between France and West Germany. But Germany still won the shootout due to two saves by national goalie, Harald 'Toni' Schumacher.
"You don't have a lot of time to react, so you need a bag of tricks," Schumacher told DW. "I wrote down everything I knew about the penalty shooters: Is he left or right footed, does he shoot hard or soft, high or low? There was a player who liked a particular direction, so I fooled him by moving my upper body just a bit in the opposite direction, knowing he'd aim at his favorite corner, and he did."
A shooter's reflexes can also provide a few clues to an observant goalie.
"I'm watching the shooter from the corner of my eye before his run up to the ball. It's only a split second, but he looks at the corner he's aiming for," said Schumacher.
The technical odds favor the penalty taker, but the psychological odds favor the goalie.
"I was never afraid because the entire burden is on the penalty taker. Since he's only 11 meters away, everyone expects him to score," said Schumacher, who was a longtime FC Cologne goalkeeper before becoming a coach for other clubs in the Bundesliga.
Germany has been unusually blessed with a succession of extraordinary goalkeepers from Sepp Maier in the 1970s to Manuel Neuer today.
Some, like Schumacher, have even had a reputation for being "penalty killers."
A good goalie can intimidate the shooter and also take some of the pressure off his teammates.
"If you know you've got a strong goalie as a back-up, you, the penalty shooter can say 'I can afford to miss'," said Schumacher.
The binominal theorem
Psychology aside, Metin Tolan has a mathematical explanation for Germany's penalty shootout success story.
Tolan's theory is based on binomial probability.
Binomial probability explains how the cumulative effect of each player's performance can improve the overall performance of the team.
In a penalty shootout, five players are chosen to score. Each individual player has an average 75 percent chance of scoring - based on factors such as the size of the goal and the goalkeeper. But the probability that all five penalty shooters score in a row is less because someone is bound to miss.
The chance of five successive goals is 75 percent to the power of five. And that equals 24 percent - which is quite low when you think about it.
Now, based on Germany's scoring average of 93 percent in penalty shootouts since 1976, let's assume that each individual player is about 10 percent better than the average player in other national teams. That would give the German players 85 percent chance of scoring a penalty.
The chance, then, that all five German penalty shooters score is 85 to the power of five. And that equals a scoring average of 44 percent. It's also pretty low - but almost double the scoring average - for five consecutive penalties - by any other average team.
In fact, the difference is about 20 percent (44 minus 24).
And this 20 percent roughly equates the 20 percent we see between the 75 percent scoring average of most teams and the 93 percent that Germany has managed in penalty shootouts since 1976.
But even after all this number crunching Metin Tolan says the math is nothing without good old fashioned practice on the field. The binomial theorem is just a mathematical tool to explain something that comes through hard work.
The Czechs practice penalties and have beaten the Germans at their own game. Antonin Panenka says that he had practiced the legendary shot that won the European Championship for his country in 1976.
"After each training session I used to stay behind with our goalkeeper and take penalties. I got the idea that if I delayed the kick and just lightly chipped it, a goalkeeper who dived to the corner could not jump back into the air. Eventually I perfected [the technique]," Panenka told UEFA, the governing body of European soccer in a statement earlier this year.
So practice makes perfect. As for luck, that also comes with practice. Golfer Gary Player's famous adage "the more I practice the luckier I get" seems to be true for the Germans and Czechs as well.
Author: Diana Fong
Editor: Zulfikar Abbany