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 28th 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…?

John_Stones_2018-06-13_1 Kyle_Walker

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 (14th 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 – 14th 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!

Thomas_Vermaelen_2018 Samuel_Umtiti_2018AUT_vs._SUI_2015-11-17_(250)

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 

Would Alien (Non-Euclidean) Geometry Break Our Brains?

The author H. P. Lovecraft often described his fictional alien worlds as having ‘Non-Euclidean Geometry’, but what exactly is this? And would it really break our brains?

 

Produced by Tom Rocks Maths intern Joe Double, with assistance from Tom Crawford. Thanks to the Oxford University Society East Kent Branch for funding the placement.

Tom Crawford, and Rockin’ Maths Matters

Esther Lafferty meets Dr Tom Crawford in the surprisingly large and leafy grounds of St Hugh’s College Oxford as the leaves begin to fall from the trees. It’s a far cry from the northern town of Warrington where he grew up.

Tom is a lecturer in maths at St Hugh’s, where, defying all ‘mathematics lecturer’ stereotypes with his football fanaticism, piercings, tattoos, and wannabe rock musician attitude, he makes maths understandable, relevant and fun.

‘It was always maths that kept me captivated,’ he explains, ‘ever since I was seven or eight. I remember clearly a moment in school where we’d been taught long multiplication and set a series of questions in the textbook: I did them all and then kept going right to the end of the book because I was enjoying it so much! It was a bit of a surprise to my teacher because I could be naughty in class during other subjects, messing around once I’d finished whatever task we’d been set, but I’ve loved numbers for as long as I can remember and I still find the same satisfaction in them now. There’s such a clarity with numbers – there’s a right or else it’s wrong. In English or History you can write an essay packed with opinion and interpretation and however fascinating it might be, there are lots of grey areas, whereas maths is very black and white. I like that.’

‘My parents both left school at sixteen for various reasons but they appreciated the value of education. My mum worked in a bank so she perhaps had an underlying interest in numbers but it wasn’t something I was aware of. I went to the local school and was lucky enough to be one of the clever children but it wasn’t until I got my GCSE results [10 A*s] that the idea of Oxford or Cambridge was suggested to me. I would never have thought to consider it otherwise.

‘I remember coming down for an interview in Oxford, at St John’s, arriving late on a Sunday night and the following morning I took a stroll around the college grounds  – I could feel the history and traditions in the old buildings and it was awesome. I really wanted to be part of everything it represented. I thought it would be so cool to study here so I was very excited when I was offered a place to read maths.

‘Studying in Oxford I found I was most interested in applied maths, the maths that underpins physics and engineering for example. ‘Pure’ maths can be very abstract whereas I prefer to be able to visualise the problems I am trying to solve and then when you work out the answer, there’s a sudden feeling when you just know it’s right.’

In his second year, Tom became interested in outreach work, volunteering to take the excitement of maths into secondary schools under the tutelage of Prof Marcus Du Sautoy OBE as one of Marcus’s Marvellous Mathematicians (or M3), a group who work to increase the public understanding of science.

‘I went to China one summer to teach sixth formers and it was great to have the freedom to talk about so many different topics. I spent another summer in an actuary’s office because I was told that was the way to make real money out of maths – it was a starkly different experience. I realised I was not at all cut out for a suit and a screen!’ Tom smiles. ‘I am a real people-person and get a real buzz from showing everyone and anyone that you can enjoy maths, and that it is interesting and relevant. I love the subject so much and I think numbers get a bad press for being dull and difficult and yet they underpin pretty much everything in the whole universe. They can explain almost everything and you’ll find maths in topics from the weather to the dinosaurs.

Take something like the circus for example – hula-hoops spinning and circles in the ring, and then the trapeze is all about trigonometry: the lengths and angles of the triangle. Those sequinned trapeze artists are working out the distances and directions they need to leap as they traverse between trapezes and its maths that stops them plummeting to the floor!’

Having spent four years in Oxford Tom then spent five years at Cambridge University looking at the flow of river water when it enters the sea, researching the fluid dynamics of air, ice and water, and conducting fieldwork in the Antarctic confined to a boat for six weeks taking various measurements in sub-zero temperatures. You’d never expect a mathematician to be storm-chasing force 11 gales in a furry-hooded parka, but to get the data needed to help to improve our predictions of climate change, that was what had to be done!

Tom also spent a year as part of a production group known as the Naked Scientists, a team of scientists, doctors and communicators whose passion is to help the general public to understand and engage with the worlds of science, technology and medicine. The skills he obtained allowed him to kick-start his own maths communication programme Tom Rocks Maths, where he brings his own enthusiasm and inspiring ideas to a new generation alongside his lectureship in maths at St Hugh’s.

A keen footballer (and a massive Manchester United fan) it’s no surprise Tom has turned his thoughts to football and as part of IF Oxford, the science and ideas festival taking over Oxford city centre in October, Tom is presenting a free interactive talk (recommend for age twelve and over) on Maths versus Sport – covering how do you take the perfect penalty kick? What is the limit of human endurance – can we predict the fastest marathon time that will ever be achieved? And over a 2km race in a rowing eight, does the rotation of the earth really make a difference? Expect to be surprised by the answers.

Esther Lafferty, OX Magazine

The original article can be found here.

Tom Rocks Maths S02 E02

The second episode of season 2 of Tom Rocks Maths on Oxide Radio – Oxford University’s student radio station. Featuring the numbers behind the sub 2-hour marathon world record attempt, P versus NP and the battle for control of the world, and the usual dose of Funbers with my super sweet 16. Plus, music from Blink 182, Billy Talent and Hollywood Undead. This is maths, but not as you know it…

Take me to your chalkboard

Is alien maths different from ours? And if it is, will they be able to understand the messages that we are sending into space? My summer intern Joe Double speaks to philosopher Professor Adrian Moore from BBC Radio 4’s ‘a history of the infinite’ to find out…

Tom Rocks Maths Episode 08

The final episode in season 1 of Tom Rocks Maths on Oxide Radio – Oxford University’s student radio station – with very special guests Jon and Nick discussing everything from the number of stickers needed to cover the Earth, to different types of infinity, via a new name for the world’s smallest number. Plus, a mammoth quiz to end the season in style and music from Nirvana and Soundgarden. This is maths, but not as you know it…

Blueprint Interview

Interview with the University of Oxford’s Blueprint magazine about my mission to popularise maths and my outreach work with the St John’s Inspire Programme. The full interview with Blueprint’s Shaunna Latchman can be found in the online magazine here.

While some avoid arithmetic at all costs, Tom fully immerses himself daily teaching maths to the first and second year undergraduate students at St Hugh’s College. He also arranges activities for St John’s College as the Access and Outreach Associate for Science, Technology, Engineering and Maths (STEM) for the Inspire programme. Another activity is planning and filming content for his own outreach programme – Tom Rocks Maths.

‘It was the subject that felt most natural to me’, explains Tom, who first realised his love for numbers aged seven when his class had been set ten long multiplication questions. He raced through the whole book. However it wasn’t until he received ten A*s in his GCSEs that he began considering an Oxbridge education. ‘Academically there isn’t much of a difference [between Oxford and Cambridge]’ Tom comments, ‘but Oxford felt more like home.’

Later, after completing his PhD in Applied Maths at Cambridge, he was offered an internship with public engagement team, the Naked Scientists. The group strip back science to help make a complicated theory easy to digest. Weekly podcasts are broadcasted through BBC Radio 5 Live and ABC Australia, where audiences reach up to one million listeners a week.

Tom saw an opportunity to bring his appreciation for maths to the masses, but he wanted to do it with a twist. Eager to move away from the stereotypes of maths being a serious subject taught by older men in tweed jackets, he thought ‘what is the best way to make maths less serious? Doing it in my underwear!’ And so, the Naked Mathematician was born.

Since joining St Hugh’s, Tom continues to break down day-to-day activities on his YouTube channel to prove that maths is an integral part of everything we do.

Tom_Crawford_2

His passion for engagement doesn’t end there. The Inspire programme, part of the Link Colleges initiative, is a series of events, visits, workshops and online contact for pupils in years 9 to 13. Tom works with the non-selective state schools in the London boroughs of Harrow and Ealing.

The Link Colleges programme simplifies communication between UK schools and the University. Every school in the country is linked with an Oxford college, with the hope that this connection will encourage students to explore the possibility of attending university.

‘The aim is to have sustained contact with the same group of students over five years,’ says Tom. ‘There are still students who haven’t thought about university, or maybe it’s not the norm in their family or area to attend university. So, we explain what it is, how it works and the positives and negatives. We want to inform and inspire them.’

Tom is responsible for arranging all STEM events across the year for 60 students in each year group. He calls on the expertise of his colleagues at Oxford as well as encouraging a partnership with the University of Cambridge and several universities in London. ‘The syllabus includes various topics such as the science of food and using maths to improve diet.’

During Tom’s famed Maths vs Sport talk, students are encouraged to participate in an on-stage penalty shootout – but only after learning about the mathematical makeup behind such a pivotal moment in a football game, of course.

Tom believes maths is made more accessible by relating it the world around us. He encourages his students to question things, like why bees make hexagonal shapes in their hives and how many Pikachus it takes to light up a lightbulb.

Whether visiting schools up and down the country to deliver talks, recording the weekly dose of Funbers for BBC radio – fun facts about numbers that we didn’t realise we secretly wanted to know – or in front of his class of students, Tom is certainly making waves in the world of maths.

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