Following Manchester City’s penalty shootout victory over Liverpool in the Community Shield, I was asked by BBC Oxford to explain how scientists are trying to find the formula for the perfect penalty…

# What are the chances that two England teammates share a Birthday?

Cast your mind back to the summer of 2018… we saw the warmest ever weather in the UK, Brexit was not *yet* a complete and utter disaster, and seemingly against all the odds the England football team reached the semi-finals of the World Cup for the first time since 1990. No doubt the team had a huge celebration together afterwards – but it wouldn’t be the first time that two of them have celebrated an occasion at the same time. As well as playing together at the heart of England’s defence, Manchester City duo Kyle Walker and John Stones also share the same birthday! Stones was born on 28^{th} May 1994, making him 24 years old; Walker was born on the same day in 1990, meaning that he is exactly four years older than his teammate. How strange! Or is it…?

On the face of it, it seems quite surprising that in an England squad of just 23 players, two of them happen to share a birthday. However, as we’re about to see, this isn’t a freakish coincidence – maths says that it’s quite likely! What we’re talking about here is commonly known as the birthday problem: if there are a group of people of a certain size, what is the likelihood that at least two of them have the same birthday?

Let’s start by saying that we have a group of N people, and assume that birthdays are equally likely on every day of the year. (There is some evidence to suggest that this isn’t the case for top athletes; some say that they tend to be born early in the school year, such as around September in England. This is because they are slightly older than the other children in the year, and so they have a slight head-start in their physical development. However, we don’t want to make things too complicated, so we’ll ignore that for now.)

The easiest way to think about the problem is to first try to work out what the probability is that *none* of the N people share a birthday. Suppose our N people walk into a room, that is empty at first, one at a time. When the first person walks in, it’s obvious that they don’t share a birthday with anyone else in the room, because there isn’t anyone else. Therefore, they have the maximum probability of not sharing a birthday with anyone else in the room, which is 1.

Now think of the second person who walks in. The only way that they could share a birthday with someone in the room is if it happens to be exactly the same day as the first person. That means there is a 1 in 365 chance that they do share a birthday, so there is a 364 in 365 chance that they don’t.

Suppose that the first two birthdays don’t match, and then the third person walks in. They now have 2 days that they can’t share a birthday with, so there are 363 possible choices out of 365. Because we assumed that the first two didn’t match, we multiply the probabilities, so now the chance that none of them share a birthday is (364/365) * (363/365).

We can repeat this process until we get to our final person, number N. For example, the fourth person has 3 birthdays that they cannot share, so we multiply by a chance of 362/365; the fifth person has 4 days to avoid, so we include a probability of 361/365… By the time the Nth person walks in, there are N-1 people already in the room, so there are N-1 days that their birthday cannot fall on. This leaves them with 365-(N-1) possibilities out of 365.

To work out the total probability, we multiply all of these terms together which gives the likelihood that none of the N people share a birthday as

**1 * (364/365) * (363/365) * (362/365) * … * ((365-(N-1))/365).**

You might be thinking that this still looks like quite a big probability that none of them share a birthday, because all of the terms are very close to 1. But, if we try some values of N in a calculator, then it tells a very different story. (The percentages are calculated by finding the probability from the equation above and multiplying by 100.)

When N = 10, we get an 88% chance that none of them share a birthday. However, this drops down to 59% when there are N = 20 people. When we get to N = 23, the number of players in the England squad, the probability reaches just under 50%. That means that, incredibly, the likelihood that at least two of the 23 people share a birthday is just bigger than 50%!

So, in a random group of 23 people, it’s more likely than not that two of them share a birthday! This seems very strange at first; surely you’d need more than 23 people for a shared birthday to be more likely than not?! This is why the problem is commonly known as the birthday paradox – it might be very hard to get your head around, but the maths doesn’t lie!

Perhaps, in order to convince ourselves, we should look at some real-life examples. This is where the World Cup squads come into play: each team is restricted to bringing 23 players to the tournament. (We’ve seen that number before…) If our calculations above are correct, then if we picked any one of the World Cup squads, there would be roughly a 50:50 chance that at least two of the squad members share a birthday, which means that out of all of the squads that went to Russia, we would expect about half of them to have a birthday match. Well, let’s take a look…

Of the 32 teams, which were divided into 8 groups of 4, the following teams have at least one pair of players who share a birthday:

Group A | Russia |

Group B | Iran, Morocco, Portugal, Spain |

Group C | Australia, France, Peru |

Group D | Croatia, Nigeria |

Group E | Brazil, Costa Rica |

Group F | Germany, South Korea |

Group G | England |

Group H | Poland |

So, not only is there at least one team in every group with a birthday match, but if we count the total, there are 16 squads with a shared birthday pair – exactly half of the teams! The experimental results have matched up with the mathematical theory to perfection. Hopefully that’s enough to convince you that our calculations were indeed sound!

A slightly different question that you might ask is as follows: if I am in a group with a certain number of people, what are the chances that at least one of them shares my birthday? Is it the same idea? What we have worked out above is the probability that *any *two people in the room share a birthday (or rather, we worked out the opposite, but we can find the right answer from our working). Note that the pair doesn’t necessarily include you; it’s a lot more likely that it’s some other pair in the group.

In order to work out the answer to this similar sounding question, we work the other way around again, by calculating the probability that none of the N people share my birthday. For each of the N people, there is only one birthday that they cannot have, and that is mine (14^{th} November, in case you were wondering), which means there are 364 out of 365 possibilities for each person. We no longer care whether their birthdays match up; we only care if they match with mine. So each person has a 364/365 chance of not sharing my birthday; and the overall probability is just 364/365 * 364/365 * … * 364/365, N times, which we write as **(364/365) ^{N}**.

Once again, we can plug some values of N into a calculator: N = 10 gives a 97% chance that no-one else has my birthday. For N = 50 the probability is still very high: there is an 87% chance that none of these 50 people have the same birthday as me. N = 100 gives 76%; N = 200 gives 58%; you have to go all the way to N = 253 before the probability dips below 50%, and it becomes more likely than not that at least one person will celebrate their birthday with me.

Applying this idea to all 736 players (32 squads of 23 players) involved in the World Cup, we should expect around 3 of them to have been born on the same day as me – 14^{th} November. And I am very happy to confirm that France’s Samuel Umtiti, Switzerland’s Roman Burki, and Belgium’s Thomas Vermaelen all have what is undoubtedly the best birthday of the year… Two similar problems with two very different solutions!

You can check which footballers share a birthday with you at www.famousbirthdays.com/date/monthDD-soccerplayer.html, where you enter the month in words and the day in numbers (no preceding zero required).

**Kai Laddiman **

# England v Colombia Penalty Shootout

I was asked by the Daily Mirror to analyse the England football team’s penalty kicks against Colombia in the World Cup second round. You can find the key insights below and the full article online here.

Image: Dr Ken Bray, University of Bath

**Harry Kane** – Kane’s very calm and confident in his walk up to the penalty spot showing that he has prepared well mentally. He carefully places the ball and adjusts his socks before firing low and hard into the bottom left-hand corner of the net. The keeper goes the right way but it’s too accurate and right in the corner of the ‘unsaveable zone’.

**Marcus Rashford** – A different approach on the walk up as he keeps his head down to make sure he doesn’t give anything away to the Colombia keeper. He curves his run-up to add extra disguise to the shot and puts it in almost exactly the same place as Harry Kane. Again, the Colombia keeper goes the right way but it’s too fast, too accurate and right in the bottom corner of the ‘unsaveable zone’.

**Jordan Henderson** – The ‘kick-ups’ on the walk to the penalty area show he’s nervous and the look on his face also hints at a lack of confidence. The placement of the shot is actually very good as he hits the ‘unsaveable zone’ to the left of the keeper, but his shot is a little higher than the previous two making it a more comfortable height for the goalie, and his wide run-up gives the game away as he opens his body to go to the right. If you look closely you’ll see that Ospina moves before Henderson kicks the ball which is why he’s able to reach beyond the ‘diving envelope’ and make the save.

**Kieran Trippier** – He has his head down and a look of complete focus on his face as he approaches the penalty spot. After a little glance up to make sure he knows where he’s going, he buries it in the top left corner in the perfect spot. Comparing Trippier’s penalty to the fourth Colombian taker, Uribe, who missed, it’s the use of the inside of his foot that makes all of the difference. Despite them both aiming for the top corner of the ‘unsaveable zone’, Uribe leant back and went with his laces making it less controlled than Trippier’s side foot. It’s also interesting that England’s nominated set piece taker went fourth in the line-up. No doubt, because Gareth Southgate knew that the fourth penalty would be key to victory as one that goalkeepers are likely to save.

**Eric Dier** – Positionally, probably the worst of the five England penalties as it was the closest to the centre of the goal and the edge of the ‘diving envelope’ which is within reach of Ospina. The key aspect of Dier’s penalty that allowed him to score was the fact that it was along the ground. Ospina dives the correct way, but can’t reach close enough to his body to make the save. Compare this to Jordan Henderson’s penalty, which was much closer to the corner, but at a more comfortable height for the save.

**Summary: **

- 4 of the 5 penalties went to the left of the goalkeeper and were all scored, whereas the one that went to the right of the keeper was saved.
- All of England’s penalty takers were right-footed.
- 2 of the 5 penalty takers were substitutes, likely brought on to take a penalty in the shootout.
- All of England’s penalties hit the ‘unsaveable zone’, maximising the chances of scoring. For Colombia only 2 of the 5 penalties hit the ‘unsaveable zone’.
- Jordan Pickford saved the fifth and final penalty, demonstrating how it is more likely for a goalkeeper to make a save later in the shootout.

England benefitted from good preparation from the manager in selecting his line-up months in advance, aiming consistently for the ‘unsaveable zone’ which is the most difficult area for the goalkeeper to reach, and by preparing well mentally and taking their time with each shot. Ultimately, these 3 things were key to the victory.

# World Cup 2018: The Perfect Penalty Kick

The 2018 World Cup in Russia kicks off today and so I bring you a special double-edition of Throwback Thursday looking at the science behind the perfect penalty kick… Fingers crossed the England players listen/read my website and we don’t lose to Germany in a penalty shootout (though let’s be honest we probably will).

Live interview with BBC Radio Cambridgeshire looking at the ‘unsaveable zone’ and the best way to mentally prepare for a penalty.

And if that wasn’t enough, here’s a full description of the ‘Penalty Kick Equation’…

For all of the footballers out there who have missed penalties recently, I thought I would explain the idea of the science behind the perfect penalty a little further, and in particular the maths equation that describes the movement of the ball. On the radio of course I couldn’t really describe the equation, so here it is:

If you’re not a mathematician it might look a little scary, but it’s really not too bad. The term on the left-hand side, *D*, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.

So, what are the inputs on the right-hand side? The symbol *p* just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that *p* will pop up in the equations – it’s sort of its job. The ball itself is represented by *R* which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by *m*. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by *r,* which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.

Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by *v*. The distance the ball has moved in the direction it is kicked is given by *x*, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is *w* – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so *w* will be small, but for Beckham – aka the king of curl- *w* will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…

So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…

# The Penalty Kick Equation

A couple of weeks ago I talked about the science behind the perfect penalty kick on BBC Radio Cambridgeshire, and lo and behold a few days later the Championship Playoff final went to penalties. I may have jinxed it – sorry. The penalty shootout was actually of a pretty high standard with 7 out of 10 penalties being scored, two saves and only one missing the target. Clearly Reading’s Liam Moore did not listen to my interview…

For all of the footballers out there who have missed penalties recently, I thought I would explain the idea of the science behind the perfect penalty a little further, and in particular the maths equation that describes the movement of the ball. On the radio of course I couldn’t really describe the equation, so here it is:

If you’re not a mathematician it might look a little scary, but it’s really not too bad. The term on the left-hand side, *D*, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.

So, what are the inputs on the right-hand side? The symbol *p* just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that *p* will pop up in the equations – it’s sort of its job. The ball itself is represented by *R* which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by *m*. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by *r,* which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.

Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by *v*. The distance the ball has moved in the direction it is kicked is given by *x*, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is *w* – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so *w* will be small, but for Beckham – aka the king of curl- *w* will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…

So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…

A version of this article was also published by the European Mathematical Society.

# The Science of the Penalty Kick

How science can help you to take a better penalty kick… Live interview with BBC Radio Cambridgeshire.

# Sub-2 Hour Marathon in Numbers

Live interview with BBC Radio Cambridgeshire on the numbers behind a sub-2 hour marathon…

You can read more about the attempt here.

# Breaking 2

The past weekend has been a big one for running: not only did Nike try and break the 2-hour barrier for a marathon; I ran my first half-marathon after a four-year hiatus due to a dodgy knee. Whilst Nike may have failed, I smashed it. Dragged around Skopje (Macedonia in case you were wondering) by my brother George, I managed to finish 25^{th} in a PB of 1:26:52 – also clocking in as the highest UK finisher. But enough about me let’s talk numbers…

By numbers I mean specifically the number 2. On Saturday morning three professional athletes, Eliud Kipchoge, Lelisa Desisa and Zersenay Tadese, set off into the darkness (literally – they started before dawn) as they attempted the impossible: a sub 2-hour marathon. For those of you that don’t run this might not mean much so let’s do some maths and break it down. A marathon is the grand old distance of 26.2 miles, also known as ridiculously far. The legend goes that a messenger Philippides ran the distance from the battlefield of Marathon to Athens to announce that the Persians had been defeated. After having exclaimed “we have won” he then collapsed and died from exhaustion, having not only run the entire distance, but also having fought in the battle himself. Legend or not (and most historians believe it to be inaccurate), I can certainly believe it. Humans are not designed to run such large distances in one go. Professional athletes we see on television make it look easy, but just think about it for a second… 26.2 miles. That’s 42.2 kilometres, 42200 metres, 105 laps of a running track – that’s insane.

But the distance isn’t the half of it. The current men’s world record marathon time is 2:02:57 held by Dennis Kimetto from Kenya and set in Berlin in 2014. That works out at 4:42 per mile… for 26.2 of them. Roger Bannister famously ran a sub 4-minute mile in 1954 (exactly 63 years to the day before Nike’s sub-2-hour marathon attempt), which is still today an incredible feat, but that is just for one mile by itself. The marathon world record pace may be a little slower, but here we are talking about 26.2 of them, in a row, with no breaks. As I said before, insane.

If you don’t know your imperial measurements (who does these days), the world record pace is 2:55 per kilometre or 70 seconds per 400m/1 lap of a running track. If I remember correctly, and I probably don’t, my best time at high school running the 400m was about 70 seconds. That’s me as a 16-year old sprinting flat out. To think that long-distance runners do that for 105 laps in a row is mind-blowing. We can continue and break it down even further into 100m segments. One marathon is equal to 422 lots of 100m, and to break the world record you would need to run each of them in a time of 17 seconds. It might not sound too hard, running a 17-second 100m race, you could probably go out and do it right now if you are physically active, but that would be one. Try doing 421 more at the same pace, in a row, with no breaks… again, insane. Hopefully you get the picture by now.

Over the past 100 years the marathon world record time has been steadily decreasing, albeit at a slower pace in recent times (as seen on the graph above). Nike’s idea was to use everything at their disposal to plan the perfect race and go sub 2 hours for the first time in history. At the level of elite athletes, taking 2:57 off a world-record is pretty much unheard of; and this attempt was not only a test of human endurance, but of science and technology too. The team behind the ‘breaking 2’ attempt consisted of world-experts in the fields of bio-mechanics, coaching, design, engineering, materials development, nutrition, sports psychology and physiology. They were meticulous in their detail, from the choice of course at Monza race track in Italy due to its favourable weather conditions, to individually designed attire for each athlete suited perfectly to their running style and technique, nothing was taken for granted. The runners were led by a group of 30-pacers that were interchanged at regular intervals to ensure energy levels were maintained. They even formed an arrow shape, with a lead man at the front, followed by a line of 2 and then 3 runners in front of the athletes, to minimise the effect of the wind.

Beginning before dawn, the three athletes set off on the journey of their lives. 17 laps of the famous circuit later, only one remained. Eliud Kipchoge completed the marathon in 2:00:25 – some two-and-a-half minutes inside the world record* but not as fast as he had hoped. It is of course unfair to label the attempt as a failure – the original BBC headline was changed from ‘Nike fail at sub-two hour marathon attempt’ to ‘Eliud Kipchoge goes close to sub-two hour marathon’’ – as how can something so incredible be seen as a failure? Kipchoge ran 26.2 miles at an average pace of 4:36 per mile – that’s only 15% slower than Roger Bannister’s 4-minute mile and he did 26.2 of them in a row!

It really was an incredible effort from one of the best athletes of our time. You could see the pain on his face as he attempted a sprint finish; this was pushing the human body to its very limits and beyond. As Kipchoge said himself after the event, the next question is how can the team take another 1 second per mile off the time? Do that and the seemingly impossible will be realised – INSANE. If that doesn’t give you goose-bumps the video below certainly will…

*To clarify the attempt does not count as a world record because of the use of pacers that were not competing in the race (along with several other infringements of the official rules).

A version of this article also appeared on the European Mathematical Society’s website.