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Dan26

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There are three doors. One door has a million dollars behind it, the other two doors have nothing.

The host of this game asks you to choose a door. You choose door number 1

The host then opens up one of the other doors (door 2) that has nothing behind it. So, this leaves two doors - door 1 and door 3.

The host says you can keep your original choice or switch to door 3

What door should you choose and why?
 

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Dan26 said:
The question is, why should you switch?
Click to expand...
Monty_tree_door1.svg


I think that covers it
 
Shell said:
Because it's an extra 33% chance you'll get the prize or something.

Was done on the movie "21", love that movie.
Click to expand...

On face value, it appears 50-50 no matter what? so what I'm looking for, is how is it possible to switch to door 3 and have a 66.6% chance of getting the million?
 
Feerits Elf said:
Because the host opened door 2 and asked if you wanted to switch.
Click to expand...

The host will always do that no matter what. No matter what door you choose, he will then always open another door with nothing behind it, leaving you with a choice of your original door, or a switch to the other.
 

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There's three possible arrangements for the doors to be sorted in:

Door 1 Door 2 Door 3 Result From Switching​
Arrangement 1 Money Nothing Nothing Nothing
Arrangement 2 Nothing Money Nothing Money
Arrangement 3 Nothing Nothing Money Money

Assuming you pick the first door every time, 2 out of the 3 situations result in you receiving the money from switching doors while you only have a single chance out of the three to obtain the money from not switching doors.

I've confused myself trying to explain what the hell is going on.

Gee, the formatting looked so nice before I pressed submit.
 
mad_monkey_j said:
There's three possible arrangements for the doors to be sorted in:

Door 1 Door 2 Door 3 Result From Switching​
Arrangement 1 Money Nothing Nothing Nothing
Arrangement 2 Nothing Money Nothing Money
Arrangement 3 Nothing Nothing Money Money

Assuming you pick the first door every time, 2 out of the 3 situations result in you receiving the money from switching doors while you only have a single chance out of the three to obtain the money from not switching doors.

I've confused myself trying to explain what the hell is going on.

Gee, the formatting looked so nice before I pressed submit.
Click to expand...

Correct.

The easiest way is to pretend there are a million doors. You choose a door, and the host then gets rid of 999,998 doors leaving just two.

There is only a one in a million chance you initially guessed correctly, which means there is a 999,999/1,000,000 chance that the money is behind the other door, even though it appears at face value to be a 50-50 chance.
 
Dan26 said:
There are three doors. One door has a million dollars behind it, the other two doors have nothing.

The host of this game asks you to choose a door. You choose door number 1

The host then opens up one of the other doors (door 2) that has nothing behind it. So, this leaves two doors - door 1 and door 3.

The host says you can keep your original choice or switch to door 3

What door should you choose and why?
Click to expand...
Is this from Brain Games?

And door 2.
 
Don't abide by this theory.

So the door that the host selects after you select yours is going to always contain a goat behind it, yes?

If that is the case, you can simply discount the hosts door from the initial thinking that the odds are 1/3 if the door the host picks is predetermined to have a goat behind it. Hence, you are virtually selecting between two doors from the start, which is of course means there is a 1/2 chance that the door you chose has the car behind it. If you change to the other door once asked by the host, you accomplish nothing as the odds are still 1/2 and comes down to random chance.
 
WaLkEr_ThE_StAr said:
Don't abide by this theory.

So the door that the host selects after you select yours is going to always contain a goat behind it, yes?

If that is the case, you can simply discount the hosts door from the initial thinking that the odds are 1/3 if the door the host picks is predetermined to have a goat behind it. Hence, you are virtually selecting between two doors from the start, which is of course means there is a 1/2 chance that the door you chose has the car behind it. If you change to the other door once asked by the host, you accomplish nothing as the odds are still 1/2 and comes down to random chance.
Click to expand...

Wrong.

There is a 1/3 chance that you select the car originally. AFTER you have selected the door, the host opens one of the doors with nothing behind it. The host will do this no matter what.

This doesn't change the fact that it was still 1 in 3, originally, and stays 1 in 3. The host isn't doing this randomly, remember. He is choosing to reveal a door that he KNOWS has nothing behind it. There is a 2/3 chance that the car was initially behind one of the two doors you didn't choose. The host gets rid of one of those doors, which means the one remaining door which you have the option of switching to has a 2/3rds chance of success.

Imagine if there were one million doors, and the host gets rid of 999,998 of them leaving just two. You still only have a one in a million chance that your original selection is the car. It's not 50-50. With a million doors initially (then down to just two), you are almost guaranteed to win the car if you choose to switch. 999,999/1,000,000 certain.

It's the same principle with 3 doors. Except it's only 66.66% certain if you switch.
 
Last edited:
Dan26 said:
Wrong.

There is a 1/3 chance that you select the car originally. AFTER you have selected the door, the host opens one of the doors with nothing behind it. The host will do this no matter what.

This doesn't change the fact that it was still 1 in 3, originally, and stays I in 3.

Imagine if there were one million doors, and the host gets rid of 999,998 of them leaving just two. You still only have a one in a million chance that your original selection is the car.
Click to expand...

I understand if you still assume the odds remain at 1/3 even after the host reveals his door has a goat behind it.

But naturally, wouldn't the odds reduce to 1/2 given you've knocked out one of the three options?

If it is predetermined that the host is ALWAYS going to open a door with a goat behind it regardless of what door you picked, you virtually had 1/2 odds of selecting the car instead of the other door with the goat behind it. If it was random chance, I was understand, but if it is predetermined, you might as well knock out the door that host is to choose from the beginning - meaning you have 1/2 chance.
 
WaLkEr_ThE_StAr said:
I understand if you still assume the odds remain at 1/3 even after the host reveals his door has a goat behind it.

But naturally, wouldn't the odds reduce to 1/2 given you've knocked out one of the three options?
Click to expand...

No.

The host has prior knowledge of where the car is. He KNOWINGLY reveals a door with nothing behind it, and will do this no matter what. There was a 2/3rds chance that the car was behind one of the doors you didn't choose. The host knowingly gets rid of one of those two doors making your job easier. It's still a 2/3rds chance that its behind one of the other two doors, but now those two doors are just one other door.

Like I said, imagine there are one million doors, and the host gets rid of 999,998 of them, leaving just two.

Given there are two doors left, do you honestly think it is 50-50? That's essentially what you are saying. Do you really think your initial selection of one-in-a-million is worth sticking with?

Or should you switch?
 
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