Hey guys, it's Bethany.
So, maybe you have heard of the Monty Hall Problem.
Well, this problem gets its name from a game show called "Let's Make a Deal" that aired
in the 60s and 70s...
The name of the games show host was Monty Hall.
So, no shame if you haven't heard of it, because it's the 21st century and cable TV is barely a thing anymore.
Well, the Monty hall problem got really famous when someone asked Marilyn vos Savant, the
smartest woman in the world, at the time, this question:
"Suppose you're on a game show, and you're given the choice of three doors.
Behind one door is a car, behind the others, goats.
You pick a door, say #1, and the host, who knows what's behind the doors, opens another
door, say #3, which has a goat.
He says to you, "Do you want to pick door #2?"
Is it to your advantage to switch your choice of doors?"
Well, she answered yes; it is to your advantage to switch doors.
So, basically in the problem that Marilyn was given,
the game show host would show the contestant 3 different doors,
and behind only one of those doors was a shiny new dream car.
Now, the contestant had to pick one door, then once the contestant chose a door Monty
would open up another door.
The door he open always had what was called a zonk behind it.
After he revealed a Zonk, the contestant then had the option to switch to the only other
door or to stay with the door that they chose originally.
At this point almost every contestant would experience high anxiety, and not know what
to do, but they would almost always stay with the door they chose first.
I mean "stick with your gut right?"
WRONG.
The problem, in this case, in sticking with your gut was that you were actually choosing lesser odds.
Here is how:
Monty says "hey girl, which door do you think your sweet new ride is behind?"
And so, the contestant chooses a door, let's say door one.
Without opening the door that you chose, Monty opens up one of the doors that you did not choose.
Since, he is the game show host he knows what is behind each door, so he always opened up
a door that the car was not behind.
This was called a Zonk, a Zonk always had something stanky behind it, like a Llama,
Alpaca, Donkey, or goat.
After he opened up one of the doors, he then asked the contestant "do you want to stay
with the door you chose, or do you want to switch.
This is when the contestants began getting really anxious, because now one of the options
have been eliminated and they know that their dream car can only be possibly be behind one
of the the two remaining doors.
Which is true, so then it would seem as though the contestant now has a 50/50 shot at getting the car.
So, like whatever.
Choose door 1 or choose door 3, who cares.
Thanks Monty for increasing my odds. Right?
WRONG.
In fact, if you switch your chance of getting the car is increased.
How?
Well, think about it like this:
if you switch, the only way that you can lose is the game
is if you chose the door with the car behind it on your first try But, remember, the odds
of choosing the door with the car behind it on the very first try are 1/3.
So, this means that your odds if you switch on the second try are .When you switch your
chances of winning are doubled.
Perhaps, you're still thinking, okay lady...
It doesn't matter if you switch.
Once, he opens the door revealing the zonk, there are only two other options, therefore
making your odds 50/50.
Boom.
Easy as that.
Get out of my face.
I will say, it is a little counterintuitive.
But, bear with me.
Let's look at this a different way.
To do this let's split the doors up into two different groups.
Group "door you chose", and group "doors you did not choose" Pretty straight forward, right?
The only door in group "door you chose" is the door you chose originally, and the doors
in the other group are the doors you did not choose.
Well, when all the doors are closed, we can easily tell that group "door you chose" has
a 1/3 chance of having the car behind it, and group "doors you did not choose", collectively,
have a 2/3 chance of having the car behind them, but each door within that group still
has a 1/3 chance.
Well, once one of the doors is opened by the host within the group "doors you did not choose"
you've now acquired a new piece of information.
You now know that the car is not behind one of the 2 doors in the "doors you did not choose group."
Well then, what happens to that 1/3 chance that went with the door that got opened in
the "doors you did not choose" group It can't just disappear into thin air.
It has to be applied somewhere.
The reason it can't just disappear is, because the total chances when winning a prize always
have to add to one.
If the 1/3 chance from the door that is now opened just disappeared, then the total odds
of winning the prize would only be out of 2/3, and that cannot be.
So, the 1/3 chance has to be applied somewhere.
You can't add it to the door in the "door you chose" group, though, because you haven't
discovered any information about that group.
You have only discovered information about the doors in the "door you did not choose" group.
So, the 1/3 chance from the door that was opened gets reassigned to the other closed
door in the "doors you did not choose" group.
So, now, the chance of winning the car by sticking with the door you chose is 1/3, and
the chance of winning the car by switching to the only door left in the "doors you did
not choose" group is 2/3.
That is the monty hall problem.
However, on the actual show "Let's Make a Deal," when the contestant chose a door,
Monty Hall never showed them another empty door and gave the contestant the opportunity to switch doors.
They chose a door, and Monty hall offered them money.
There was none of this switching business.
Marilyn Von Savant simply proposed the situation in a magazine article at the time, and the
problem became known, from that as The Monty Hall problem...
but in no way reflects what actually happens on the show.
I hope that this video has been helpful.
If you enjoyed it, be sure to give us a like, and subscribe to our channel to check out
more of our videos.