OK, so today we’ve got exclusive

access to Nottingham Forest football ground here. Two-time European Cup winners

who have given us the access so that we can talk about a

very important number in football, and a very important

number in probability, which is the number 23. OK, so 23. So why is 23 an important

number in football? Some of you may know

this already. That’s the number of people

that you will have on the pitch during the game. That’s two teams of 11

and the referee. So there are 23 people

on the pitch. Now, here’s my question today. What’s the probability that two

of those people will share a birthday? The answer may surprise you. So we’re not talking about the

year, we’re just talking about the date itself. Maybe it’s the 14th

of January, maybe it’s the 5th of June. We’re talking about

the date itself. So let’s work this out. I can make this slightly

easier if I ask the opposite question. I’m going to work out the

opposite, which is, what’s the probability that no one on that

pitch shares a birthday. That’s an easier question

to answer. Let’s do this. OK, so your first player. OK, it doesn’t matter what

birthday he has. But when your second player gets

on the pitch, what’s the probability that he doesn’t

share a birthday? Well, he will have, out of the

365 days to choose from, he can have a birthday on 364

of them out of 365. So we’re not including February

the 29th, no leap years here. And we are assuming that all

days are equally likely. OK, so your second player has

to have one of these days. Your third player, when he comes

on to the pitch, will have a choice of 363

days out of 365. And then what next? The fourth player. Out of those remaining days, he

will have 362 out of 365. And you can keep going. Eventually, you’ll get

to the 23rd player. Let’s call him the referee. So your 23rd player, how many

choices does he have? He will eventually get 343

days left out of 365. So this is the number of days

that you’re allowed to have for that referee’s birthday. Because we are looking at no

one sharing a birthday. Now, if you want to find out

the probability that no one shares a birthday, you multiply

all these together. And you’ll get a number. And that number is around

about 0.493. And if you’re not happy with

probabilities like that, that’s 49.3%. Just slightly under half. We were interested in the

opposite question. The opposite question was,

what’s the probability that someone does share a birthday. That’s the opposite thing of

what we’ve worked out. So the probability that someone

does share a birthday will be 50.7%. It’s slightly over a half. You’re more likely for two

players to share a birthday than if they don’t

share a birthday. And that’s quite surprising. And people want to think,

well it must be something like 100 people. You must need 100 people

for that to be true. Or 200 people. If you think of it this way,

think of all the pairs of people you could make out of

23 people on the pitch. All the possible pairs of

people– in fact there are 253 pairs of people you

could make. And, well, think of it that

way– you start to see why it’s quite likely, that

two of those people will share a birthday. So next time you’re at a

football match, think of it– you have a greater than 50%

chance that two of those people share a birthday. OK, so here’s another way

to think about it. Imagine you’re watching

the game. And well, if you need to go to

the toilet, you get up and you have to move past all

the other people. By the time you’ve moved past 23

other spectators, there is a greater than 50% chance that

two of those people you walked past will have shared

a birthday.