23 and Football Birthdays - Numberphile

Uploaded by numberphile on 14.02.2012


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.