23 and Football Birthdays – Numberphile


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.

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