Game Show

[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 - who knows where the car is so as not to end the game prematurely - 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?

Note:

The location of the car can not be changed.

The host's only motive for not revealing the car when he opens a curtain- in this case, curtain #3 - is solely so he can give the contestant a chance to switch his choice.

 

[Stephen]

On the face of it, it's simple. You have a 50/50 chance of choosing the correct curtain. So why change?
However the intervention of the game show host changes the psychology.
You say the host opened curtain #3 so as not to end the game prematurely.
But are you either mistaken or lying about his/her motive?
Was that the real motive or was it to try and induce the contestant to switch because the host knew the car was behind curtain #1?

Either way I'd stick with curtain #1.

 

[tango]

This is one of those probability questions where the addition of the new information doesn't seem to change things but actually does.

I'd need to refresh my memory and run the numbers again to remember exactly how it works but it's something like you have a 50/50 chance of winning if you stick but a 66% chance of winning if you change.

 

[JRT]

Regardless of anything that went on before. the contestant knows that the car is behind one of two curtains. So whether he switches his choice or not it is a 50-50 choice.

 

[tango]

Regardless of anything that went on before. the contestant knows that the car is behind one of two curtains. So whether he switches his choice or not it is a 50-50 choice.

That's how it looks at face value. Some years ago a friend with a PhD in mathematics explained why it's not that simple. I had to think about it to remember how it works, and it's something like this:

If the game were to view three curtains, watch the host open a curtain to reveal no prize, and then pick a curtain to see whether you win, you'd have a 50% chance of getting the prize. But the choice followed by the host opening a curtain changes things.

Based on three curtains and one prize you have a 1/3 chance of being right and a 2/3 chance of being wrong. If you picked the correct curtain the host can open either of the other two, and changing your mind means you lose the prize you might have won had you stuck with your first choice (this outcome has a 1/3 probability). If you picked one of the curtains that does not conceal the prize the host is forced to open the other empty curtain, and therefore changing means you win the prize (this outcome has a 2/3 probability).

Hence changing your selection increases your chances of winning to 2/3.

 

[Stephen]

That's how it looks at face value. Some years ago a friend with a PhD in mathematics explained why it's not that simple. I had to think about it to remember how it works, and it's something like this:

If the game were to view three curtains, watch the host open a curtain to reveal no prize, and then pick a curtain to see whether you win, you'd have a 50% chance of getting the prize. But the choice followed by the host opening a curtain changes things.

Based on three curtains and one prize you have a 1/3 chance of being right and a 2/3 chance of being wrong. If you picked the correct curtain the host can open either of the other two, and changing your mind means you lose the prize you might have won had you stuck with your first choice (this outcome has a 1/3 probability). If you picked one of the curtains that does not conceal the prize the host is forced to open the other empty curtain, and therefore changing means you win the prize (this outcome has a 2/3 probability).

Hence changing your selection increases your chances of winning to 2/3.

Your analysis might have merit IF the host was obliged to open a curtain as part of the game. But the OP says he did it to prolong the game implying his decision to do so was optional, or at least dependent on you picking a curtain with the car behind..

 

[Lamb]

There's a wiki page for this:

Monty Hall problem - Wikipedia

en.wikipedia.org

 

[Stephen]

There's a wiki page for this:

Monty Hall problem - Wikipedia

en.wikipedia.org

That is interesting but rstrats version is not identical to Savant's (apart from the goats)
In Savants there was a rule that "The host must always open a door to reveal a goat and never the car."
But in rstrats' version the opening of a door appears to be optional since the host is said to open it to prolong the game not because he/she was obliged to.

 

[tango]

Your analysis might have merit IF the host was obliged to open a curtain as part of the game. But the OP says he did it to prolong the game implying his decision to do so was optional, or at least dependent on you picking a curtain with the car behind..

Even if the host wasn't obliged to open another curtain the numbers stay the same.

When you took your original guess you had a 2/3 chance of being wrong, and the absence of the car behind the curtain the host chose to open doesn't change that. Whether the host was required to open a curtain or not doesn't change anything - the fact is that the host did open a curtain and then invite you to change your selection.

Feel free to explain how it changes the numbers.

 

[Stephen]

Even if the host wasn't obliged to open another curtain the numbers stay the same.

When you took your original guess you had a 2/3 chance of being wrong, and the absence of the car behind the curtain the host chose to open doesn't change that. Whether the host was required to open a curtain or not doesn't change anything - the fact is that the host did open a curtain and then invite you to change your selection.

Feel free to explain how it changes the numbers.

I did in post #2

 

[tango]

I did in post #2

I wouldn't call that an explanation.

 

[Stephen]

I wouldn't call that an explanation.

I agree with you that if the rules are fixed, as in Savant's case then the odds are calculable and the best option is to switch.

But, as I interpret it, in rstrats' version the host has some discretion. If that is the case then human psychology comes into it.
Did the host really open the curtain just to prolong the game, or was he/she trying to induce the contestant to switch because he/she knew the contestant had picked the one with the car?

 

[rstrats]

Stephen,
re: "I agree with you that if the rules are fixed..."

They are.


re: "...and the best option is to switch."

It is.

 

[tango]

I agree with you that if the rules are fixed, as in Savant's case then the odds are calculable and the best option is to switch.

But, as I interpret it, in rstrats' version the host has some discretion. If that is the case then human psychology comes into it.
Did the host really open the curtain just to prolong the game, or was he/she trying to induce the contestant to switch because he/she knew the contestant had picked the one with the car?

If you really want to go there you could also argue that the people running the show knew you got it right and took the chance to move the car behind the other curtain though while the host was making a big show of opening a different curtain, right?

 

[Stephen]

If you really want to go there you could also argue that the people running the show knew you got it right and took the chance to move the car behind the other curtain though while the host was making a big show of opening a different curtain, right?

I disagree. To switch the car would be cheating. It would be totally dishonest.
In either example I don't see that the host is cheating or being being dishonest.

 

[tango]

I disagree. To switch the car would be cheating. It would be totally dishonest.
In either example I don't see that the host is cheating or being being dishonest.

Why would it be cheating? The rules as listed don't say the car can't be moved during the game.

 

[Stephen]

Why would it be cheating? The rules as listed don't say the car can't be moved during the game.

OK, your silliness beats my logic.

 

[rstrats]

The rules as listed don't say the car can't be moved during the game.

They do now.

 

[tango]

OK, your silliness beats my logic.

I see. So if you point out the rules don't say something then it's logic but if I do it then it's silliness? Glad we got that cleared up.