The Monty Hall problem is based
upon an old game show,
where a contestant picks one of three doors.
Behind one door is a new car.
Behind the two other doors are goats.
For this video, though, we will be changing the prizes.
There is a prize of $1 million behind one
of three doors.
There is $0 behind the other two doors.
You pick one of three doors.
Then, one of the two doors you did not pick is opened
that has $0 behind it.
You are then given the choice to keep your door choice
or switch to the other remaining unopened door.
Should you keep your original choice,
or should you switch doors?
Does it matter?
Well, it actually does matter.
Let's take a closer look at why it matters.
As an example, let's say you select door one.
There is a 1/3 chance the $1 million is
behind door one,
which also means there is a 2/3 chance
the million dollars is behind door two or door three,
and now if you are shown there are $0 behind door two,
the question is should you keep your first choice
of door one,
or should you switch your choice to door three?
Well, because there's a 2/3 chance the million dollars is
behind door two or door three,
and now you know the million dollars is not behind door two,
there is now a 2/3 chance the million dollars is
behind door three,
which means you should switch doors.
Theoretically, by switching doors you have a 2/3
or approximately 66.7% chance of winning $1 million.
By keeping the original door choice, you have a 1/3
or approximately 33.3% chance of winning $1 million.
To better understand this, let's take a look
at a similar situation with more doors.
Once again, there is a prize of $1 million
behind now one of 10 doors.
There is $0 behind the other nine doors.
You pick one of the 10 doors.
Then eight of the nine doors you did not pick are opened.
All eight have $0 behind them.
You are then given the choice to keep your door choice
or to switch to the other remaining unopened door.
Again, should you keep your original choice,
or should you switch stores?
So again, let's say you select door one.
There is a 1/10 chance the million dollars is
behind door one.
There is a 9/10 chance the million dollars is
behind a door numbered from two through 10,
and now if you are shown there are $0 behind door two
through door nine,
again, the question is should you switch your choice
to door 10 or keep your original choice of door one.
Because there's a 9/10 chance the million dollars is
behind a door numbered from two through 10
and now you know door two
through door nine don't have the million dollars,
there is now a 9/10 chance the million dollars is
behind door 10,
which means, once again, you should switch doors.
Again, theoretically by switching doors you have a 9/10
or 90% chance of winning $1 million.
By keeping the original door choice,
you have a 1/10 or 10% chance of winning $1 million.
One last thing I do want to mention before we go is
that the Monty Hall problem is not the same
as someone first showing you one door that has $0,
let's say this door here, then making your first choice
and then being given the chance to switch doors.
This would be the same
as having just two doors to choose from.
If this was the case,
switching doors would not increase the probability
of winning.
There would be a 1/2 or 50% chance of winning regardless
of the door chosen or if you switched doors.
Thank you for watching.