Monty Hall Three-Door Puzzle

Solution 1:

The reasoning "because there are two doors, the probability that the prize is behind either is $1/2$" depends on an unspoken premise that there is no reason to distinguish between the two doors.

But the situation is not really that symmetric. One of the two doors were chosen by the contestant without any special information. The other was chosen by the host, under the particular restriction that it and the contestant's choice must not both be non-winning.

Because the two doors were not chosen under the same conditions, the situation is not symmetric between them, and hence the argument that they should be ascribed the same chance of winning is not valid.

Solution 2:

Here's the way I like to think about it. When you choose the door, you have 1/3 chance of being right. If you have the opportunity to choose 2 doors, you have 2/3 chance of being right. By offering you to switch, Monty effectively offers you to choose 2 doors rather than 1 (the two doors are the closed one, and the one he has already opened). Thus, you have 2/3 chance of being right if you switch.

Extra: Many people like to give the multiple doors example. Let's say that you have 100 doors instead of 3. You choose one of them and then Monty opens the other 98 doors, leaving one closed and giving you the chance to switch. Most people would feel that there's something fishy going on with that door — why didn't he open that specific door? So most people would now intuitively switch.

Solution 3:

could someone clarify or correct my reasoning?

i've found the concept behind mh problem is clear when changing the scale. for example, instead of three doors, say, use $100$ doors. $99$ goats are behind 99 doors and the car is behind a single door. obviously you have a $\frac{1}{100}$ chance of picking the car. in other words, the odds are slim you picked the winning door. now monty, who knows where the car is located, opens the $98$ doors that all hide goats, leaving your door, which you choose with $\frac{1}{100}$ chance of succeeding, and one other door. he asks you to pick again. clearly your odds on the second attempt are much better than $\frac{1}{100}$.

what is at fault with my reasoning? the crux of my argument depends upon the $98$ doors being revealed before your second choice.

-j

Solution 4:

I like to think of the host combining the two boxes that the contestant has not picked and saying that the contestant can walk away with the contents of both the boxes that were not picked. Almost equivalent if the losing boxes are empty and with negligible difference in the goat variation (though I daresay it matters to the goat).

Solution 5:

Remember that this is a game as well as a probability puzzle. It's always nice to have the right tools. I had trouble with the puzzle until I made the connection.

Very simply, a game has a state and some rules. The state begins in a specific condition, that is, there are three doors with one car and two zonks, but you don't know which door has the car. Monte does know, so we call this imperfect information (an example of perfect information is chess, where both sides can see all the pieces).

You get to make the first move, by picking a door. Monte makes the second move, by opening one of the other two doors. The state has changed. Not only that, you have information you didn't have before. You've learned that one door is definitely car-free.

Here is the the real puzzle. You've also learned something about what Monte knows. Because Monty knows where the car is, he is restricted in the moves he can make, that is, he can't open the door with the car or the door you picked. Initially, the probability of your door being correct is 1/3 because they're all 1/3. After Monte's move, the probability for your door is still 1/3, because Monte isn't allowed to change its state even if it has the car. Meanwhile, the open door's probability has dropped to 0, so the last door's is 2/3. To Monte, at this point, the correct door is 0 (because you win) and the other is 1.

Now we get into psychology. Monte offers you the chance to switch. Because of the Endowment Effect, you feel that you already own your door. You can see that there are only two doors, so they must be equally valid choices. There's no real reason to accept his offer--except that both these reasons are misleading.

In the context of Let's Make a Deal, it gets more interesting. If you stay with (or switch to) the correct door, Monte knows you're about to get a car. To him, that's a loss. He can offer to trade your door (to you an unknown quantity) for that box over there, or whatever's in this envelope, or a weekend trip to Vegas (still worth less than the car). He can't give every contestant a car, but he'll raise suspicion if he doesn't give away any. In effect he's cheating by giving himself more moves.