Hello Ding-a-lings, my name is Michael and today we’re going to discuss the Monty Hall
Problem.
But why?
Why should anyone discuss the Monty Hall Problem.
It has been discussed ad nauseam.
There are so many articles and videos about it that are fantastic including Kevin’s
recent video about paradoxes on Vsauce 2.
Check that one out, please.
Does the world really need another discussion of the Monty Hall Problem?
No.
But maybe it does.
You see I believe that true understanding comes from listening to as many voices as
possible, discuss something from as many perspectives as possible and so the more people you listen
to talk about the Monty Hall Problem and why it’s solution is what it is, I believe through
osmosis the greater your understanding will be and lately for the last few weeks I have
been driving around on my commute to and from the Vsauce office talking out loud to myself
ab out the Monty Hall Problem.
If you see me talking to myself in my car I’m not on the phone.
I’m literally explaining the Monty Hall Problem to myself, honing in on the best way
to intuitively get it and I feel like I have found an explanation that…well that I like.
I’m not saying it’s the best but I wanted to share it with you in the spirit of understanding.
So let’s dive in.
First of all what is the Monty Hall problem.
Well it was named after game show host, Monty Hall who hosted let’s make a deal.
One of the challenges he would put the audience to involved three doors on the stage: door
number one, door number 2, and door number 3.
Now Monty would honestly tell the audience member that behind two of the three doors
was a goat.
A goat.
Yeah.
But behind one of the doors was a bunch of money.
Let’s say a million dollars.
Now the audience member got to pick one of the three doors and if they picked the door
that had money behind it they would get to keep the money but if they picked a door that
had a goat behind it, well they’d get a goat and they’d have to like take care of
it and travel back home with it.
Point is, you weren’t supposed to want the goat but goats are awesome she I, in my version,
have replaced the goats with pieces of poop.
Fewer people want poop.
Some still will but the point is you’re supposed to want the money.
If you pick a door that has poop behind it you have to, like the goat, take the poop
home, and take care of it and feed it, all that normal poop stuff.
So here we go.
Uhh door one.
Door one.
Okay.
Awesome.
Before I open door one, I’m gonna tell you this.
It’s good that you picked door number one because there’s poop behind door number
two.
You see that?
There’s some poop.
Oh you don’t want that.
But now, Hannah before I give you what’s behind the door that you have chosen I’m
gonna actually let you switch.
You can switch to door number three or you can stick with your original choice.
After you’ve done that, either you’ve switched or not, you will have to take home
whatever is behind the door that you have chosen.
Would you like to switch or stick with your original choice?
I’ll stick with my choice.
Why?
Because it doesn’t matter.
It doesn’t matter.
It doesn’t matter right?
It kinda feels like a 50/50 right.
You know that there’s no money behind door number two which means the money’s either
behind door number one or door number three.
So you have a 50/50 chance right?
I mean why not just stick with your first choice because what if your first choice was
right and ya switched.
Well then you would really hate yourself for not going with your gut.
It would feel like you lost the money.
Like you had it and then you lost it by switching.
If it’s behind the door you didn’t pick then it’s like you just picked wrong.
You don’t feel like you’re losing something so psychologically yes most people decided
not to switch.
But mathematically you should always switch.
In fact if you switch you will win not half the time, it’s not actually 50/50.
If you switch you will win 2/3 of the time.
And if you don’t switch you’ll only win 1/3 of the time.
So you have a 66.6 repeating chance of winning if you switch.
Why should that be?
Why is it not 50/50?
Well it comes down to two extremely important facts about this problem.
The first one is quite easy for us all to agree on.
It’s the fact that when you make your first choice, oh by the way I never actually showed
you the result.
You stuck with 1, and Hannah just like I said, you should have switched.
You shouldn’t always switch, let’s be clear about that.
I’m not telling you a strategy to always win.
I’m just telling you the best strategy in the long term.
So you chose to stick with one but there’s poop behind one.
The money was behind three.
So there it is, a million dollars to scale.
So, let’s get back to these two important facts.
The first is that when you first choose a door the probability that you have chosen
the money is 1/3.
Okay that’s the chance that you’ll be right.
The money is behind one of three doors.
You have a one in three chance of picking the door with money behind it right off the
bat.
That’s pretty clear but that also means 2/3 of the time you are choosing a door that
does not have money behind it.
More doors have poop behind them than have money behind them.
The second and much more important fact is that the host knows which door the money is
behind and the host will never open a door that has money behind it.
I believe most disagreements over the solution of the Monty Hall Problem come from confusion
over the rules of the game.
So let’s make it extremely clear.
First you pick a door.
Next the host opens one of the two doors you did not pick.
And the door the host opens of those two will always be a door with the goat behind it.
You are then asked if you would like to stick with your original choice or switch to the
door that you didn’t pick that the host also didn’t open.
This is why it feels like a 50/50.
We know that the money is either behind the door you picked or the door that the host
didn’t open that you didn’t pick.
It’s behind one of them because it’s definitely not behind the one the host opened.
The host only ever opens a door with a goat behind it.
But surprisingly to switch or not is not a 50/50 choice.
Switching allows you to win 2 out of 3 times.
If you chose a door like Hannah did, door number one say, and then the host flipped
a coin and randomly opened one, sometimes the host would open a door that had the money
behind it and that would be terrible television because the money would be revealed and the
host would say okay well you were wrong, would you like to switch to the one door that remains
closed.
Of course you wouldn’t.
It wouldn’t matter.
The game would be over and it would be over.
Instead the host always opens a door of the two you haven’t chosen that has poop behind
it.
Okay.
Or a goat depending on which version you wanna go with.
So what does that mean?
Well it means that sometimes you choose the door that the money is behind.
That’s awesome.
And in that case it doesn’t matter which door the host opens.
The host could have a rule where they always open the left most door or maybe they just
flip a coin and open one or two.
Either way, the host can open either door.
But most of the time you didn’t pick the door the money is behind.
2/3 of the time you pick incorrectly.
In that case the host is forced to open one door.
The host has no choice.
The host cannot open three because the money is behind that door.
The host must open the other door to reveal the poop.
This is why you actually have more information than you might think.
The host is always avoiding the door with the money behind it.
So by looking at which door when you have the choice to switch is still closed.
You actually know a little bit more about what might be behind that door.
In the case that you picked correctly the first time, you picked a door with the money,
the host has decided to open one door and not the other simply because of random chance.
But most of the time you’re choosing incorrectly and the host has left a door closed.
The door that the host left closed was left closed because there’s money behind it and
you should switch.
Okay.
To make this really really clear, if I choose correctly, the door that remains closed has
no money behind it.
If I don’t choose correctly the door that remains closed does.
So if I’m wrong with my first choice I should switch.
If I’m correct with my first choice I shouldn’t switch.
Okay?
Perfect.
Here’s the thing.
You choose correctly much less often if we were to run many many trials you choose correctly
much less often than you choose incorrectly because you’re only picking the door with
the money behind it 1/3 of the time.
2/3 of the time, most of the time, you will have chosen a door with poop or goat, maybe
it’s goat poop behind it.
So most of the time you should switch because the door that remains closed is the money
door.
There’s an analogy that I think makes this even more clear.
And it involves a sack.
This sack contains 3 marbles.
Two are white and one is black.
We can say, let me put them on the bag, great.
Okay now we can say that the goats are represented by the white marbles and the million dollars
are the black marble.
You want the black marble and when things begin I don’t know which marble is which.
I just have to reach in there and pull one out and after I’ve pulled on out I don't
get to look.
Perfect.
I’ve pulled one out.
I don’t know if this is a white or black marble but I want the black one.
Remember that.
But now what happens, an analogy to the Monty Hall problem with money and goats is that
I give this bag to the host.
The host then looks inside the bag and pulls out a white marble.
The host will always pull out a white marble because the host will never reveal the black
marble.
The host will never open the door that has the money behind it and there will always
be at least one white marble left in the sack because if I chose the black marble then there
are two white marbles in here.
If I chose the white marble then there’s a white and a black.
Either way the host has a white marble to remove.
The question now is should I switch?
Should I go with the marble I initially picked?
I don’t know what it is.
Or should I switch tot eh bag?
I should always switch to the bag because I’m much more likely to have chosen a white
marble.
The host then removes the other white and there’s a black marble in here.
If I choose correctly with my draw that means I’ve pulled out the black marble and there
are two white marbles in here.
Well then of course when the host removes one white marble there’s still a white in
the bag and I shouldn’t switch but I choose the black marble initially only 1/3 of the
time.
Most of the time I am handing the host a bag that contains the black marble and one white
marble.
The host’s job then is to remove a white marble, leaving most of the time the black
marble that I want in the bag and so I should switch to what’s in the bag and not what’s
in my hand.
Now you should be prepared to make a killing on let’s make a deal.
Very good.
Stay curious, keep exploring probability and as always thanks for watching.
Oh yeah!
Oh I'm grr baby grr.
Get it?
It's Borat.