By Andrew Puopolo

Today, Robert Lewandowski came on as a halftime substitute for Bayern Munich as they trailed 1-0 at home to VfL Wolfsburg. 14 minutes later, following the most incredible sequence of events, Lewandowski had scored 5 goals and Bayern Munich led 5-1 after goals in the 51^{st}, 53^{rd}, 55^{th}, 57^{th} and 60^{th} minute. Clearly, this feat has never been achieved before on such a high level and I wondered, what is the probability of this occurring? What is the probability that Lewandowski could score 5 goals in 10 minutes?

To easily find the probability of Lewandowski achieving this feat, I calculated his minutes per goal ratio in Bundesliga matches since he broke onto the scene in 2011/12 when he scored 22 goals for Borussia Dortmund. His minutes per goals ratio can be interpreted as the probability that he scores in any given minute of the game. Since 2011/12 and before Tuesday, Lewandowski had scored 86 goals in 11,140 minutes of Bundesliga action. I used his Bundesliga scoring ratio and not his Champions League/Europa League scoring ratio because the Bundesliga is a uniform competition (one match against each team in the league at home and one away) with an almost uniform level of competition. Also, because his goal scoring ratio for Dortmund and Bayern Munich is almost identical (.56 goals a game for Dortmund and .57 goals a game for Munich), it is OK to use the larger sample size as the opposition is still constant. We can call his goals to minutes ratio *p* and let that represent the probability that Lewandowski scores in any given minute of a Bundesliga match. To express the probability that he scores five or more goals in a ten minute stretch, we can use the following binomial distribution:

N.B. I assumed that the minutes in which he scored the goals to be distinct to simplify calculations. Also, it is nearly impossible to score twice in the same minute of a soccer game given that it takes a lot of time to celebrate the goal, run back to the halfway line and let the other team kick off before stealing the ball marching down the field and scoring again.

In this representation, *p *represents the probability that Lewandowski scores in any given minute, so for each minute he scored in we must multiply our expression by *p*, and for each minute that he did not score in by *(1-p)*, as that is the probability that he does not score in any given minute. The choose function represents the number of ways in which we can choose the distinct minutes in which Lewandowski scored. So if he scored 5 goals, then there are 10 C 5 ways to determine the exact 5 minutes in which Lewandowski scored. By default, he did not score in the other five minutes.

Substituting our value 86/11140 in for p, we get that the binomial distribution evaluates to or to put in fractional form: 1/1.5×10^8

This is equivalent to *one out of 150 million.* However, we must multiply our result by 80 because there are 80 such 10 minute stretches in any given game so actually the probability he scores 5 goals in 10 minutes is 1 out 1.875 million. Now, to figure out how often we would expect this to happen, we take this probability and divide it by the number of games in a Bundesliga season (34). Doing this calculation tells us that a player of Lewandowski’s caliber would be expected to accomplish this feat once every 55,147 years.* *

If we wanted to determine the probability that a team in general scores 5 goals in 10 minutes, we can use a team (lets say Bayern Munich in this example) and use the same formula to calculate the probability of the whole team scoring 5 times in 10 minutes (like Germany against Brazil in the 2014 World Cup). Since Bayern Munich (a very proficient scoring team) have scored 430 goals in the last 5 Bundesliga seaons (a total of 15,300) we can express their goals per game ratio *p *as 43/1530.n Plugging that value into our binomial distribution above yields a value of or 1/254744. So the probability that a team like Bayern Munich (who are one of the most efficient scoring teams in Europe) scores 5 goals in 10 minutes is 1 out of 250,000. However, once again we must multiply by 80 because there are 80 such stretches in a game and we find that this has probability 1/3184. Since there are 34 games in a Bundesliga season, we can expect a historic club like Bayern Munich to score 5 goals in 10 minutes to happen once every 94 years.

To roughly determine the probability of this event taking place in one of Europe’s top 4 leagues (La Liga, Bundesliga, the Premier League and Serie A), I calculated the number of players who scored more goals than Lewandowski (17) in those leagues last season. This list contains Cristiano Ronaldo, Lionel Messi, Antoine Griezmann, Neymar, Carlos Bacca and Aritz Adruiz from La Liga; Alexander Meier from the Bundesliga (Arjen Robben also scored 17); Sergio Aguero, Harry Kane, Diego Costa and Charlie Austin from the Premier League and Mauro Icardi, Lucas Toni, Carlos Tevez and Gonzalo Higuain from Serie A. In total, there were 15 players who scored more than Lewandowski last season. (Note: The Bundesliga only plays 34 matches while the other three leagues play 38, but to keep the calculations simple and understandable I used this methodology). To center the probability around Lewandowski goals per minute probability, I also took into account the 15 players who scored less than Lewandowski. Since Lewandowski is expected to achieve this feat once every 55,147 years, I divided this by the 31 best players in Europe to determine that this event is likely to take place in one of the top 4 leagues in Europe once every 1,778 years.

So basically, if Lewandowski scored 5 goals on the day Jesus was born, this would not be expected to happen again until Benjamin Franklin signed the Declaration of Independence.

Astounding.

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