The simplest children in the streets have long known that one should switch doors when faced with the Monty Hall problem. A variant is proposed as follows:
Monty Hall presents three doors to Xavier, you, and Zebediah. Two doors conceal a goat, and one door conceals a car (assume that all contestants prefer a car to a goat). Each contestant chooses a door and communicates this choice to Monty without revealing this choice to the other contestants; if more than one contestant chooses the same door, all contestants choosing that door are awarded an instance of the prize behind it.
After each contestant has chosen a door, Monty reveals that each door has been chosen by exactly one person. He then opens the door chosen by Xavier, and there is a goat behind it. Monty offers you and Zebediah the choice to keep your respective doors or switch (again, if you both wind up with the same door you each get an instance of the prize behind it).
Should you stick or switch? Should Zebediah stick or switch?
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5 comments:
There's a good chance I could be reading the question wrong, but it doesn't appear that the number of contestents should have any affect on your choice. It's like saying, door #1 had a goat, but the Dallas Cowboys just lost. Do you want to switch your choice?
I admit that it seems weird that it is in Zeb's best interest to pick your door, and for you to pick Zeb's door, but the math still seems to work out that you, and Zeb, have a 2/3 percent chance of getting the car if you switch.
It can't be the case that you both have a 2/3 chance of success by switching:
To say that you have a 2/3 chance of getting the car if you switch is equivalent to saying that there's a 2/3 chance that the car is behind Zeb's door. Then clearly Zeb would have only a 1/3, not a 2/3, chance of success by switching.
And vice versa, mutatis mutandis, etc.
I guess the main point of debate is if you have a 2/3 chance of getting the car if you switch or a 1/2 chance of getting it if you switch. Regardless, you should always switch, (aka it can't hurt you).
I've always held that the probability of winning is 2/3 if you switch, but reading the wiki article, the probability can be 50% if the host is clueless which door the car is behind, aka he/she might reveal the car instead of the goat. While it wasn't mentioned in the writeup, I assume that in this scenario the host won't reveal the car.
It might be significant though that if you had picked the goat originally, the host could have revealed your door with a 50% probability, (unless he/she really doesn't like Xavier...)
Doing the truth table on it, and throwing out the results where the host reveals you have a goat, shows you winning 50% of the time when you switch. This seems to match up with what Rasalom is saying.
I guess the key then is to be freindly to the host, or hope that Xavier is a bore. Conversly if the host caught you sleeping with his wife before the show and he didn't reveal you had the goat, you should stick with your original pick ;)
As in the one-contestant problem, the correct answer depends on unstated assumptions. For example, if Monty is a random agent, you don't get any information about what's behind closed curtains from his actions. The one-contestant problem depends on the assumptions that Monty knows where the car is and will always heighten the drama by revealing an unselected curtain which hid a goat.
With three contestants, the second assumption does not obtain, since there are no unselected curtains. Monty has two choices of curtains that conceal goats, but he will be spoiling the chances of one contestant. How does he choose which one?
Assume Monty always reveals a goat, and that he uses some random process to determine which one (with 50% probability of choosing each one).
It does seem puzzling at first why the odds should be different from the case of the classical problem - it seems like you could just ignore Zebediah and it reverts to the one contestant case. I also found it easier to understand by doing a kind of truth table.
I think Matt put his finger on the reason why the odds change - over many iterations, 1/3 of the time it's going to be your door that's revealed and Xavier and Zeb who need to decide to stick or switch.
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