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?
You should change your choice to curtain 2. Although you make think it's actually a 50/50 call, you are actually wrong. You only had a one in three chance of guessing right in the first pace and a two in three chance of it being curtain 2 or 3 and the game host conveniantly eliminates curtain 3 for you meaning there is a 2 in 3 chance of the car being behind curtain 2. If you have trouble following my reasoning imagine a hundred curtains and the game host progressively eliminating 98 of the remaing 99 curtains.
A bit of a random question, but yeah AA is correct. Intuitively we'd want to say that we'd stay with our original guess as nothing has changed, but if you mathematically do the math, once the 3rd curtain is unveiled your first guess was almost certainly wrong.
Actually, I think it might be best to go for curtain #1. If the game show host wanted to extend the show by opening a different curtain than #1, his reason was probably that the car was behind curtain #1. Else he would have had no reason the open #3 instead of #1 to make the game longer, knowing the car was behind #1, the one first chosen, and therefore cutting the show short if it was exposed on the first guess. Had curtain #2 or #3 been chosen first, he would have had no reason to fear the game ending too soon, so would simply opened which ever one was actually chosen.
Tiffy, re: "Actually, I think it might be best to go for curtain #1. If the game show host wanted to extend the show by opening a different curtain than #1, his reason was probably that the car was behind curtain #1. " That is incorrect. His reason for not opening the contestant's initial pick is always to give the contestant the opportunity to change to a different curtain.
Then you should have stated that at the beginning, not half way through the problem. What is the point of the contestant choosing a curtain if the game-show host then breaks the rules of the game and opens a different curtain, (ostensibly to play for time. The only reason given in the puzzle for him doing so). Logically the only reason the game-show host would 'play for time', would be if the car was behind curtain #1 and the contestant had chosen curtain #1. Stupid game anyhow, if the game-show host can open curtains the contestant didn't pick. What is the point of that? Ahh! I know. So you can wind people up by always telling them it's the wrong answer. Which cup is the pea under? Place your bets ladies and gent's. No one will be a winner, because the pea is now in my pocket.
I generally find that sticking with one's first pick is best. I'm not a math person, but it seems to me that even if there was only a 1/3 chance when the contestant initially picked curtain #1 that it was right, the odds of it being right don't remain so when the third curtain is eliminated. Because there is an opportunity to choose after the elimination, the choice is now 50/50. Given that curtain #1 survived the elimination once, I believe you should always dance with the date that brung ya. Trying to argue that a a 50/50 choice is some how slanted in one favor or the other doesn't seem to hold water statistically speaking. Alternatively, I heard a story about how some Arab oil tycoons used to break casinos in the 80s by using the simple formula of playing a number/ color over and over again, doubling the amount that was put down every time it lost until it finally won. They had some evidence that everytime you lost, the odds of you winning on the same number/color ob the next play went up versus playing another number. From what I understand, when Las Vegas heard they were coming, it was so scared that it shut down all the casinos. That story may or may not be true, but I think it points to staying with curtain #1.
Lowly Layman, re: "I'm not a math person, but it seems to me that even if there was only a 1/3 chance when the contestant initially picked curtain #1 that it was right, the odds of it being right don't remain so when the third curtain is eliminated. Because there is an opportunity to choose after the elimination, the choice is now 50/50." What if after you had selected curtain #1, and before any curtains were opened, the host told you that you could switch to both curtains #2 and #3. What would you do?