Three Curtains

[rstrats]

You are a contestant on a game show. There are three curtains. Behind one of the curtains is a new car. You are asked to choose one of the curtains. Lets say that you choose curtain #1. The host of the show opens curtain #3 and there is no car behind it. The host now gives you a choice. You can stay with curtain #1 or you can change your choice to curtain #2. The question now is: would it be to your advantage to stay with curtain #1, or would it be to your advantage to change to curtain #2 or would there be no advantage either way?

[Eeset-Shadowgrl]

I have always found more gain in sticking with my intuition. Therefore I would stay with my original choice.

[Ben Zwycky]

Logically you improve your chances by changing your mind, it's called the 'Monty Hall problem', I believe.

1. To start with you have a 1/3 chance of being right, and a 2/3 chance of being wrong,

We could put it like this:

P(first guess is right) = 0.3333, P(first guess is wrong)= 0.6667

When the host opens curtain 3, he is giving you additional information since he knows which curtain the prize is really behind, he is reducing the risk of choosing P(first guess is wrong), the probability of which hasn't changed.

If you think your first guess is wrong (which is more likely than not) and the host didn't show you anything, then you still don't know which of the other two curtains to choose, so you'd still have an additional 50% chance of guessing wrong there, meaning the individual probability for either of those options is

0.6667*0.5=0.3333, i.e. all options have the same probability.

Once he does tell you which curtain it is not behind, this 50% chance falls away and you are left with

P (first guess was right)= 0.3333 vs P(first guess was wrong)= 0.6667,

so you double your chances by changing your mind.

Another explanation by way of examples:

Case 1: The prize is behind curtain 1

You choose curtain 1
Host reveals curtain 3 is empty
You change your mind to curtain 2

Result: You lose!

Case 2: The prize is behind curtain 2

You choose curtain 1
Host reveals curtain 3 is empty
You change your mind to curtain 2

Result: You win!

Case 3: The prize is behind curtain 3

You choose curtain 1
Host reveals curtain 2 is empty (because he knows which curtain the prize is behind)
You change your mind to curtain 3

Result: You win!

So in two out of the three possible cases, changing your mind wins you the prize, in one out of three cases it loses you the prize, so your chances are doubled by changing your mind.

[rstrats]

Here are a couple of other ways to explain it:

What if after your initial pick of curtain #1, I told you that you could stay with #1 or switch to BOTH #2 and #3? My guess is that you would switch so that you could look behind both curtains in spite of the fact that you KNOW that at least one of the curtains does not have a car behind it. Lets say that the first curtain that you open is #3 and you find that there is no car behind it. You now get to open curtain # 2, giving you 2 chances. What is the difference if you open curtain #3 or the host opens curtain #3 for you? Either way, you get to look behind curtain # 2.

You can also think of it as two areas. Area "A" contains curtain #1 and area "B" contains curtains #2 and #3. There is a 1/3rd chance that the car is in area "A" and a 2/3rds chance that it is in area "B". Before opening any curtains, you KNOW that at least one of the curtains in area "B" doesn't have the car behind it. So by opening a curtain in area "B" that doesn't have a car behind it doesn't change the 2/3rds odds that area "B" still has a car in it.


I wonder, though, if it would make any difference to the probabilities if the host DIDN’T know where the car was? Naturally it would end the game prematurely if he opened the car curtain, but if by chance he didn’t, would there still be a 2 to 1 improvement by switching?

[Ben Zwycky]

No, it doesn't change anything, if it doesn't end the game early, because it would still remove the 50-50 chance from changing your mind, basically you're betting on getting the answer wrong first time (which is a good bet probability-wise)

[Moksha]

You are a contestant on a game show. There are three curtains. Behind one of the curtains is a new car. You are asked to choose one of the curtains. Lets say that you choose curtain #1. The host of the show opens curtain #3 and there is no car behind it. The host now gives you a choice. You can stay with curtain #1 or you can change your choice to curtain #2. The question now is: would it be to your advantage to stay with curtain #1, or would it be to your advantage to change to curtain #2 or would there be no advantage either way?

No advantage either way.

[Ben Zwycky]

Um, I've just shown above that you double your chances of success by changing your mind

[Moksha]

Um, I've just shown above that you double your chances of success by changing your mind

Yes I saw that but I don't believe the "logic". (Unless I'm not understanding the original question.)
As I understand it you start of with a one in three chance of picking right. Right?
You choose curtain 1.
Host reveals curtain 3 to be empty.
Asks you if you want to change your mind.
Ok so now you have a 1 in 2 chance of being right. Yes?
It doesn't matter if you stick with 1 or change to 2 it's still a 50/50 chance of being right.

[Ben Zwycky]

If there were only two curtains to start with, then you would have a 50/50 chance of being right.

You have a one in three chance of being right with your first guess, and two in three chance of being wrong with your first guess. When the host shows you the empty curtain and asks whether you want to change your mind, he is effectively asking:

Do you want to bet on being right with your first guess (one in three chance) or wrong with your first guess (two in three chance)?


Does that make more sense now?

[Moksha]

If there were only two curtains to start with, then you would have a 50/50 chance of being right.

You have a one in three chance of being right with your first guess, and two in three chance of being wrong with your first guess. When the host shows you the empty curtain and asks whether you want to change your mind, he is effectively asking:

Do you want to bet on being right with your first guess (one in three chance) or wrong with your first guess (two in three chance)?


Does that make more sense now?

The arguement is convincing I just have a hard time believing that it's true. It sounds like one of those lying with statistics/numbers kind of things.

Do you want to bet on being right with your first guess (one in three chance) or wrong with your first guess (two in three chance)?

In this part you are saying it's a choice between (1 in 3) and (2 in 3) which sounds sensible. But still it's were you right or wrong with your first guess. (1 in 2)

(I might have to do the experiment to convince myself)

[Ben Zwycky]

It's two possibilities, yes (right or wrong), but one of them is more likely than the other, there was only one way to be right with your first guess and two ways to be wrong.

OK, how about a different example of the same principle to make things clearer, say there were a hundred boxes to choose from, you pick one at random, then the host opens 98 of the others to show they are empty and asks you if you want to change your mind.

Don't you agree that it is much more likely that you were wrong the first time, so changing your mind (effectively choosing the 99) would be a good idea?

[rstrats]

Change the sentence in the OP which reads: “The host of the show opens curtain #3 and there is no car behind it” to “The host of the show - who knows where the car is so as not to end the game prematurely - opens curtain #3 and there is no car behind it.”

[rstrats]

Moksha,

re: "(I might have to do the experiment myself).”

To that end you might want to take a look at the following link:

http://www.nytimes.com/2008/04/08/sc...nty.html?_r=2#

[Moksha]

Moksha,

re: "(I might have to do the experiment myself).”

To that end you might want to take a look at the following link:

http://www.nytimes.com/2008/04/08/sc...nty.html?_r=2#

Thanks all.
I think my head is begining to get itself around this now. I guess I just needed that experiential input.