Marilyn vos Savant. She's an American.
She writes a regular newspaper column.
And she's the world's most intelligent person. She has the highest IQ in the world.
And one week she showed her readers a puzzle about a game show.
[Game show music plays]
OK, who wants to come down and play "Don't Get the Goat"?
Young man at the back there, that's it. Stand just over there.
What's your name? - Jeremy.
Jeremy. OK, Jeremy, you're going to play "Don't Get the Goat".
I'll explain it first.
Behind these three doors, we have three prizes.
Behind one of them is a booby prize of a goat.
Behind the second one here.
Behind the third one there, is another booby prize, another goat.
But behind one of the doors is star prize of a red sports car.
OK, it's a bit easy at the moment, so we're going to shut them up again.
I'm going to shuffle them around.
Jeremy here is going to face away, look at the audience.
And all the kind people over here are going to make sure that
although they can see exactly what I'm doing, they don't tell Jeremy where the car is.
OK, shuffle them around.
Great.
OK, you can look now, Jeremy.
Now...
I want you to choose one of the doors but don't open it yet.
Choose a door, any door.
That one?
OK, that's your choice, so I'm going to mark it like that.
Now I'm the game show host, I know where the goats are.
And I'm going to open one of the doors and show you that
that one has got a goat.
Now, you happy with that choice or would you rather choose this door?
Do you want to change your mind?
Happy with that one.
You're happy with that one.
OK. Let's see whether you were right.
[Game show jingle]
Yes!
So that time the contestant stuck with their choice. They got it right.
But if we kept doing that, was that the correct strategy?
Should Jeremy actually have done that?
Well, now what Marilyn told her readers was -
in the long run that's the wrong thing to do.
It worked on that occasion, that's probability, but it won't always work.
And here's her argument.
The car could be behind any one of these three doors.
Equal probabilities, one in three.
The probability it's behind that door is one in three.
So when the contestant chooses that door,
there is a one chance in three that's the correct door.
Two chances in three that the car is behind one of the other two doors.
Now when the game show host opens a door to show a goat,
he knows where the goat is.
If the car is in there, he's not going to open that door.
So there's still a two thirds chance that the car's behind one of these two doors
and it's not that one so it must be that one.
So what Marilyn said was - there's a 1 in 3 chance it's behind the door you first chose
and then because of what the host does,
there is a 2 in 3 chance it's behind the other door.
So you should change your mind.
And yet you look at this and say, hold it, there's two doors there.
Isn't it equally likely that the car would be behind that door or behind that door?
What's the point in changing my mind, it can't possibly make any difference?
Nevertheless, Marilyn was right.
I'm going to try and convince you of this, because it's not at all intuitive.
So to do this, I'm going to exaggerate the whole thing.
I need another volunteer to come around here and help me.
Young lady there, you want to come around and help?
OK.
Come here, stand in here.
If we're going to exaggerate this, we're going to have nine goats, represented as cards.
And one car, OK?
Do you want to just check? Lots and lots of goats, one car.
So I'm going to shuffle these.
I'll shuffle them away.
And then I'm going to lay them out.
And I want you to try to choose the card that's got the car,
just slide it out there, don't turn it over yet.
Choose a card, any card.
One of them might be the car. Pull it out.
Which one? That one?
OK, there's your choice. What's your name by the way?
Claire. - Claire.
OK, so Claire's chosen that one.
Here's the rest of them.
OK, Claire, what's the chance that the car is that card do you think?
There's ten cards altogether. - 1 in 10.
1 in 10. Absolutely right. 1 in 10. 1 time out of 10, that's the car.
So 9 times out of 10 the car is in here somewhere.
So if at this stage I said to you - alright, pick one of those piles
and if the car's in that pile, you win, which pile do you want?
That one. - That one.
Right. OK?
Change your mind.
I'm the host, I know some of these cards are goats.
So I can go through here and I can discard a whole load of goats
and I can put the one that was left down there.
So now, which pile do you prefer?
That one.
Absolutely right.
OK, Claire, thank you.
We look at two cards at the end, just like two doors,
and we say, unless we're careful, the car is equally likely to be behind these.
But you saw from the way those cards were prepared that
there it's a 1 in 10 chance, there...
I very carefully selected and played around,
so there's a nine tenths chance it's in here.
Now, for the game show we saw, it's the same thing,
but it's as if there are two goat cards and one car.
So in fact Marilyn is correct, the chances are not equal for the two doors you're looking
at.
And if you change your mind, in the long run, you win twice as often as if you don't.