Hello, everyone!
Today I'm going to do a modern classic for you,
or, to put it another way,
a modern cliché,
which is why I want to get it out of the way.
Today we're going to do
"The Monty Hall Problem--"
the Monty Hall Problem.
Now, the Monty Hall Problem is very famous these days.
The idea is this:
We're on a game show,
and I'm your game show host--
you're going to have to imagine
the sparkly jacket--
and we've got three doors.
Now, behind one of these doors
is a car,
and behind each of the other two doors
is a booby prize,
which for some strange, inexplicable reason
is a goat.
Now, our budget doesn't stretch
to a car and two live goats,
so we're going to have to play for something else.
We're going to have to
play for
this Mogwai.
So that's what we're playing for today.
We'll just--we'll put him down.
He doesn't like the bright lights.
So we'll put him down there.
Now,
you're going to pick a door.
So let's say you pick door number 1.
Now Monty Hall, your game show host,
will reveal one of the booby prizes.
So let's say door number 3--
so there's nothing behind door number 3.
OK, now you're left wiih a choice
of two doors.
And that's what Monty Hall is going to do.
He's going to say, "Do you want to stick with your original choice,
or do you want to switch?"
Now you might be thinking, "Well, there's only two doors--
it's 50/50. It makes no difference
if I stick or switch."
In fact, you are twice as likely to win
if you change your mind.
Now, this is very strange, and it's counter-intuitive.
This problem was famously posed in a 1990 edition of Parade Magazine.
And thousands of people wrote in, saying it's 50/50.
Even mathematicians wrote in saying it's 50/50.
But it's just counter-intuitive.
A quick appeal to intuition might be to imagine 100 doors.
So I've got 100 doors,
and I'm going to ask you to pick a number between 1 and 100.
Then I'm going to open 98 of those doors,
leaving the door you picked
and one other door.
Now, how likely were you
to pick the right door to begin with?
Do you think it's more likely to be
behind your door,
or the other door that I left unopened?
Now, with three doors,
it becomes a bit more counter-intuitive.
Here we go!
There are three options.
You can have the car behind the first door,
and a goat behind the second door--a goat behind the third door.
In the second case,
you could have goat-car-goat,
and in the third case,
you would have goat-goat-car.
So let's say you pick door number 1,
And in the first case,
If you stick with door number 1, you're going to win.
So let's put a tick next to that.
And Monty Hall--he's going to open, say, door number 3, or door number 2,
to reveal the booby prize,
and if you switch, and change your mind, you're going to lose,
so we'll put an x there.
But in the second case,
if I stick with door number 1,
I'm going to lose,
so let's put an x next to that.
And Monty Hall--well, he has to open door number 3--
he's forced to open door number 3,
which leaves the car behind.
So if you change your mind, you're going to win the car.
So let's put a tick next to that.
And in the third case,
if you stick with door number 1, you're going to lose.
So we'll put an x there.
And if you change your mind,
you're going to win the car.
So we'll put a tick there.
And, as you can see,
if you stick with your original choice,
you win one out of three times.
But if you change your mind,
you will win two out of three times.
So the reason this fools you
is because it's not genuinely random.
If it was genuinely random,
it would look something like this:
Monty Hall would open, say, door number 3,
without knowing what's behind it.
So he might accidentally reveal the car.
If he opens door number 3 and reveals a goat,
That means we're not in that third case,
so we can get rid of those,
and that's what we're left.
And as you can see,
it becomes 50/50,
if you change your mind or not.
This is how the modern equivalent of Deal or No Deal would work.
The Mogwai was behind door number 2.