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 No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?
Here I don’t want to approach the problem statement from statistics perspective, but from a human perspective.
I think that the answer to the problem above depends on answer to following question:
Is the host trying to fuck with you (knowing about the Monty Hall problem)?
If the show is on for a long time and he’s opening a goat door every show / the game rules dictate him to always offer the switch, then the best option is switching.
If however it’s the first week of the show / or the door is opened seemingly randomly, it makes sense to reconsider the options and how smart of a person the host thinks you are.
The additional assumptions stated in the article do solve this issue (by trying to eliminate the human factor):
Although not explicitly stated in this version, solutions are almost always based on the additional assumptions that the car is initially equally likely to be behind each door and that the host must open a door showing a goat, must randomly choose which door to open if both hide goats, and must make the offer to switch.
OK, indeed there is “a fully unambiguous, mathematically explicit version of the standard problem is” mentioned in the article.
Suppose you’re on a game show and you’re given the choice of three doors [and will win what is behind the chosen door]. Behind one door is a car; behind the others, goats [unwanted booby prizes]. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one [uniformly] at random. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?
- If we want to go nerdy about the “Behind one door is a car; behind the others, goats [unwanted booby prizes]” -it does not state that behind each of the other doors, there are ONLY goats – so feel free to assume one of the other doors has “a goat and a car”, or it could be both / one of the goats being statues internally filled with gold.
- And what if the player has motorphobia?
P.S. The explanation of the solution I do like the most:
As Cecil Adams puts it (Adams 1990), “Monty is saying in effect: you can keep your one door or you can have the other two doors.”