Hello
Today I'm going to be talking
about the Birthday Problem,
sometimes called the Birthday Paradox,
and this is how it goes:
so let's suppose you have a room
and unrelated people start entering the room
and you ask them to start
comparing their birthdays.
You imagine that most of the time
when two people compare
there's a very small chance
of them matching birthdays.
Most of the time they're not going to match.
But if you get a large enough
number of people in the room
you imagine
there probably is a chance that eventually
you'll find two that match.
So here's the question.
How many people do you need in a room
before you have a 50:50 chance,
a 50% chance,
of two people sharing a birthday?
So maybe you think it's about 50?
But then again 50 only represents
less than one seventh
of the possible number of birthdays
so maybe you want a few more than that?
What about 100?
But even with 100 there's still
more than two thirds of the possible birthdays left.
So maybe you want
half the number of possible birthdays?
There are 366 possible birthdays
if you include February 29th
so we want 183 people maybe?
Or maybe you think even more than that?
Maybe 300?!
So I'll give you a couple of seconds to think about
how many
you need in the room
and then I'll tell you what the answer is.
OK so here is the answer.
The number of people you need in a room
before you've got a 50:50 chance
of two of them having the same birthday
is ...
23!
Yes, just 23.
So it's quite a counterintuitive result
if you've never seen this before
just how few people you need
but before you say I'm
just telling lies and it's all wrong
we're going to prove it using probability.
So when we're doing this proof
we're going to make some assumptions.
The first is that
February 29th doesn't exist.
Now obviously it does exist
but it makes the maths a little bit harder.
So it doesn't really change the result very much
so we're going to ignore it for the moment.
Second we're going to assume that
each birthday is
equally likely to come up.
So we're going to ignore
slight increases in certain days
and whatever else.
And lastly we're going to assume that
each of the people in that room
are independent from each other.
So we're not talking about a
convention for twins or anything like that.
The importance of the independence condition
is that we can start multiplying
[the probability of] events together.
Right. So to prove this
we usually phrase the opposite question.
So instead of how many people before
two people share the same birthday,
what's the probability
that no-one shares a birthday?
I.e. that every single birthday
in that room is unique.
So obviously when one person enters a room
You can't have a match
because there aren't enough people yet.
When the second person enters the room
to not match it must avoid
the birthday of the first person
who is already in the room.
So there are 364 possible birthdays
that will do that
so the chance is 364 in 365.
When the third person enters the room
they must avoid two birthdays
so now there's only 363 days left
so the chance of the third person
avoiding both birthdays
is 363 in 365
and then to get the probability that
none of them share a birthday
we multiply it
by the result we had from the previous stage.
When the fourth person enters the room
it's 362 in 365
times what we had before.
When the fifth person enters the room
we're trying to avoid four birthdays now
so it's 361 in 365
times what we had before
and each successive time
we need to avoid one more birthday.
By the time we have nine people in the room
the probability that everyone
avoids each other's birthdays
has dropped to about 90%
This means that
the opposite condition:
the chance that at least one pair
shares a birthday
has nearly risen to 10%.
By the time we have 15
people in the room together
the probability that everyone
avoids each other's birthdays
has dropped to less than 75%.
This means there's a greater than
25% chance,
1 in 4,
that two of them will share a birthday
with only 15 people in the room.
And as the 23rd person enters the room
the probability
has now dropped to 49.27%.
So that means the chance of two people
having a birthday together
is now above 50%.
So yes this really means
that you only need 23 people
to be in a room together
before there's a 50% chance
that two of them have the same birthday.
So next time you're in a room
with a group of people
why not try and find out if two of those people
have the same birthday?
In fact it works in all sorts of situations.
So in fact
by the time this video
has had its 23rd view
there's a 50:50 chance
that two of those viewers
have the same birthday
and actually it ramps up really quickly
by the time this video has had
57 views
there's a 99% chance
that two of them somewhere
will have the same birthday.
If it gets 100 views
there's a 1 in 3 million chance
that no-one shares a birthday.
So a really tiny chance!
So thanks for watching!
