Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors? (The assumption is that you'd prefer to win the car )
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Depends on how well you've got on with the host. If you think you've got on really well I'd follow through on his leading question and pick door number 2. I would have preferred a choice as follows: It seems fairer. There are 3 doors marked with the signs. Goat Goat + 3 wheeled Car to travel with Goat Car (a real beauty) None of the doors has what the signs says : You are allowed to open only ONE doors and from this guess what is behind the other doors. The host opens the third door and if all doors has what it says on the signs you switched you win the car. That seems fairer! (if it works....... ) regards [ November 11, 2003: Message edited by: HS Thomas ]
As the problem is stated, it is ABSOLUTLY to your advantage to switch. Every time. This is actually one of the most debated questions ever in Marilyn Vos Savant's column history. Think of it this way. Suppose there were 1,000,000 doors, one with a car, and all the rest having goats. you pick one. odds are pretty good you picked a goat. Now, Moty Hall opens 999,998 doors - leaving the ONE you didn't pick, and ONE more. you gonna swtich now? the same logic applies to 3 doors. here's another approach. there's a black goat, a white goat, and a car. case: switch assume you initially picked the black goat. Monty open the white goat door, you switch, you win. If you initially picked the white goat, monty shows you the black, you switch, you win. you originally picked the car, monty shows you a goat, you switch, you lose. so you win 2/3 times. case: Don't switch pick the black goat. shown white goat. lose. pick the white goat. shown black goat. lose. pick car, shown a goat, win. win 1/3 times.
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As the problem is stated, it is ABSOLUTLY to your advantage to switch. Every time. No, it's ambiguous. This is actually one of the most debated questions ever in Marilyn Vos Savant's column history. Primarily because MVS didn't specify the problem well enough to remove ambiguities, and subsequently failed to recognize equally valid answers from her readers. Bert also didn't specify the problem unambiguously, but I suspect he'll do better the MVS at recognizing this. The key issue actually is: when the host opens a door for you, does he do that because he always does that, i.e. because it's part of the rules of the game? Or is it something that he chose to do, based on his knowledge of the situation, and on his own motivation? It's probably a given that he wouldn't choose to open the door which actually has the car behind it - but the question is, could the host decide not to open a door at all if it suited him? If the host always opens a door, and the contestant konws that, then it's to the contestant's interest to switch, as Fred argues above. But if opening the door is a choice by the host, then the question isn't really answerable, unless we know exactly what motivates the host - how does he choose whether or not to open a door? [ November 11, 2003: Message edited by: Jim Yingst ]
Of course, if the host does not open another door, there is no reason for him to ask you if you want to switch doors. As long as you find out that there's a goat behind door #3 after choosing one of the other doors (either the host opend the door, you overheard a stage hand whisper that it was a goat, you looked with your x-ray vision and saw the goat, whatever...), then it is advantageous for you to change door.
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Of course, if the host does not open another door, there is no reason for him to ask you if you want to switch doors. But it may be that sometimes he opens a door, and asks if the contestant wants to switch, and other times he doesn't open a door, and therefore doesn't ask the contestant if they want to switch. What if the host follows an algorithm like this:
So if the host likes you, it's to your advantage to switch, since that choice would only be offered to you if you hadn't already picked the correct door. But if he doesn't like you, he offers the second choice only if you did pick the correct door, and he's now trying to trick you. So it's not necessarily in your interest to switch when offered the option, unless you know the host's motivation. Or, unless the host always opens a second door and offeres a switch, in which case you do have statisticlaly better chances if you switch, always. [ November 11, 2003: Message edited by: Jim Yingst ]