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The main reason Monty Hall is non-intuitive is that the shift in probability is rather small (relatively. To make it more intuitive, we can simply make the shi
by tomp 11y ago
The main reason Monty Hall is non-intuitive is that the shift in probability is rather small (relatively.
To make it more intuitive, we can simply make the shift in probability absurdly high!
Imagine you have 1000 doors, behind one there is a car and behind others, there are goats. You pick one door. The show host opens all other 998 doors, except yours and another door, revealing only goats. Do you think it's more likely the car is behind your door or behind the other unclosed door?
- zdkl 11y agoP('My Door') = P('Any other Door')?
- purplelobster 11y agoThe problem is in the phrasing, and you're actually not improving on it. The way you explain the problem, it would still be 50/50. What changes the situation is that the show host CANNOT open the door with the car in it. The way you phrase it, it's not clear that the show host has any information that the participant doesn't have, but he does.
- tomp 11y agoYes, I guess making that explicit would make it even more obvious.
- benmmurphy 11y agoHe does kind of improve on it. You do have information about whether the host knows which door the car is behind. If the host does not know which door the car is behind he just did something very unlikely so you should update your model based on this information. Assuming you have reasonable priors about the distribution of whether the host has information about the car then after he did what he did it is very likely he has information about where the car is. Though actually I don't think you can then use this information to answer the question because it is tainted with the assumption that the car is not behind your door. :(
- purplelobster 11y agoYes, but that's not what this problem is about. Monty Hall actually knew where the car was and never picked it, that's how the show worked. The problem is that whenever the Monty Hall problem comes up, this is never clear in the phrasing. For someone who have never seen "Let's make a deal", this is not obvious, and thus people are confounded by the problem when really it's not very confounding if you knew the facts. On a side note, I think what you talked about would be 0.2% chance: product((n-1)/n) n=3 to 1000 I'm not a stats/probability wiz, but I suppose if you need to decide between Monty Hall using his knowledge or not, you'd be fairly certain by this point...
- lmm 11y agoIt still depends on what the show host is doing. Maybe he only opens 998 other doors if you happened to pick the right one in the first place. To a monkey brain that's more likely than him trying to help you get the car.
- maxerickson 11y agoAt the time it was first popularized, there was no need to speculate about what the show host was doing. Statements of it often leave out explicit discussion of the broader context of the problem, but saying "Monty Hall" relates it directly back to a game show where the host always revealed a goat.
- lmm 11y agoThe report I read was that he didn't always do it in the real gameshow.
- maxerickson 11y agoIf he opens a door and shows the car, the game is over with nothing to win (or they get the car?). If he doesn't open a door, it isn't the Monty Hall problem.
- lmm 11y agoIf he had the option of not opening a door and just opening the one the contestant picked immediately, then that makes it a rather different problem from the mathematical one. And AIUI that was how the gameshow worked: sometimes Monty would reveal a goat and offer you the chance to switch doors, and sometimes he wouldn't.
- mistermann 11y agoThis is the very best technique of explaining the Monty Hall problem I've ever read.