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 of course 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?
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Three Curtains
- Thread starter rstrats
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TOG
Member
Since you've already seen that there is no car behind curtain #3, it becomes irrelevant to the equation. The car is behind one of two curtains, and without any further information, there is a 50% chance for each. There is therefore no advantage either way, whether you change your mind or not. If I were in that situation, I would base my choice on the host's behavior. He wants to save money for the show, and doesn't want to give the car away. If he seems to be trying to get me to change my mind, I would stick with curtain #1, but if he seems to want me to stick with my first choice, then I'd change it. But based purely on mathematical probabilities, there's the same chance for both curtains.
That was the correct solution. I imagine that the one you want to hear is probably somewhat different and goes something like this...
Since you have 3 curtains, there is a 2/3 chance of the car not being behind the curtain you chose. Since you've ruled out curtain #3, there is a 2/3 chance that it is behind curtain #2, and therefore it would pay to change your mind. But like I pointed out, once it's been opened, curtain #3 becomes irrelevant to the calculations.
That was the correct solution. I imagine that the one you want to hear is probably somewhat different and goes something like this...
Since you have 3 curtains, there is a 2/3 chance of the car not being behind the curtain you chose. Since you've ruled out curtain #3, there is a 2/3 chance that it is behind curtain #2, and therefore it would pay to change your mind. But like I pointed out, once it's been opened, curtain #3 becomes irrelevant to the calculations.
The TOG
TOG,
re: "...based purely on mathematical probabilities, there's the same chance for both curtains."
What would you do if after your initial pick of curtain #1, and before any curtain was opened, the host told you that you could switch to BOTH curtains 2 and 3? And the host is not trying to trick or influence you in anyway. Even If the car is behind curtain 2 or 3, he will still give you the offer to switch.
re: "...based purely on mathematical probabilities, there's the same chance for both curtains."
What would you do if after your initial pick of curtain #1, and before any curtain was opened, the host told you that you could switch to BOTH curtains 2 and 3? And the host is not trying to trick or influence you in anyway. Even If the car is behind curtain 2 or 3, he will still give you the offer to switch.
TOG
Member
You didn't mention anything about who opens the curtains. Besides, that's irrelevant. The difference is that in the first case you mentioned, one curtain had been opened before I was asked whether to change my mind, and the choice was between two curtains. In the second example, the choice was not which of two curtains I wanted, but whether I wanted to pick one curtain or two. The two scenarios are not comparable.
The TOG
TOG,
re: "You didn't mention anything about who opens the curtains."
But I did. The OP tells you that after your pick of curtain #1 that the host opens curtain #3. When you are given the chance to switch to curtain # 2 you say that there would be no advantage in doing that. Yet you say that you would switch to curtain #2 and curtain #3 if given the chance before curtain #3 is opened. So I ask you again - what is the difference if you personally open both curtains or the host opens curtain #3 for you?
re: "You didn't mention anything about who opens the curtains."
But I did. The OP tells you that after your pick of curtain #1 that the host opens curtain #3. When you are given the chance to switch to curtain # 2 you say that there would be no advantage in doing that. Yet you say that you would switch to curtain #2 and curtain #3 if given the chance before curtain #3 is opened. So I ask you again - what is the difference if you personally open both curtains or the host opens curtain #3 for you?
TOG
Member
I've already answered that. It's obviously better to pick two curtains than just one. If you pick just one, you have a 1/3 chance of picking the right one, but if you pick two, you have a 2/3 chance of being right.
The TOG
TOG,
OK, let me put it a bit differently. You say that you would switch to curtain #2 and curtain #3 if given the chance before curtain #3 is opened. So you walk over to those 2 curtains. The host is there, also. The host suggests that you open one of the curtains, and that he opens one of the curtains. If the car is behind either curtain it's yours. You agree and the host goes first and opens curtain #3 and of course there is no car behind it. Now you take your turn and open curtain #2. How is that any different than if you were to switch when initially given the chance that was offered in the OP?. You
OK, let me put it a bit differently. You say that you would switch to curtain #2 and curtain #3 if given the chance before curtain #3 is opened. So you walk over to those 2 curtains. The host is there, also. The host suggests that you open one of the curtains, and that he opens one of the curtains. If the car is behind either curtain it's yours. You agree and the host goes first and opens curtain #3 and of course there is no car behind it. Now you take your turn and open curtain #2. How is that any different than if you were to switch when initially given the chance that was offered in the OP?. You
TOG
Member
I get the feeling you are trying to confuse us. You have no presented 3 different scenarios and seem to be trying to get us to think that they are all alike.
Scenario 1
Number of curtains excluded: 1 (#3)
Number of curtains to choose: 1
Number of curtains to choose from: 2 (#1 or #2)
Chances of having chosen correctly: 1/2
Scenario 2
Number of curtains excluded: 0
Number of curtains to choose: 1 or 2
Number of curtains to choose from: 3
chances of having chosen correctly: 1/3 or 2/3, depending on what is chosen
Scenario 3
Number of curtains excluded: 0, then 1 after a decision has been made
Number of curtains to choose: 1 or 2
Number of curtains to choose from: 3
Chances of having chosen correctly: 1/3 or 2/3 depending on what is chosen, but 1/2 after one has been revealed to be wrong.
The TOG
Eugene,
re: "Regardless the spin anyone puts on it, the odds went down once the host revealed that one of the curtains didn't have the car from two-thirds to one-half."
That is incorrect. 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 the host knowingly opening a curtain in area "B" that doesn't have a car behind it doesn't change the 2/3rds probability that area "B" still has a car in it. So it would be to your advantage to switch to the remaining curtain in area "B".
re: "Regardless the spin anyone puts on it, the odds went down once the host revealed that one of the curtains didn't have the car from two-thirds to one-half."
That is incorrect. 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 the host knowingly opening a curtain in area "B" that doesn't have a car behind it doesn't change the 2/3rds probability that area "B" still has a car in it. So it would be to your advantage to switch to the remaining curtain in area "B".
Sounds great if the car isn't in Area A.
I dunno. What if curtains #1 and #3 is area B instead of curtains #2 and #3?
