[javajosh]

Dammit you beat me to it! Although maybe I can contribute because I was going to state it thusly...

Imagine a related Monty Hall problem, where you select a door, and then Monty immediately asks (without revealing anything), "Do you want to keep that door, or would you like to pick the other two doors?" Clearly you'd pick two doors instead of one.

When monty opens a door and gives you the "choice to switch" he is making noise designed to make picking two doors look like picking one door.

[beernutz]

Thank you. When you put it that way, it makes so much more sense to me. Opening the door does not change the odds. Why is that so counter intuitive?

[javajosh]

Glad it helped! Why it's counter-intuitive-ness is a really, really good question because like 99.9% of people I found this one hard to accept, too.

All I can say is that it's an incredibly effective verbal/logical obfuscation that relies on synergistic choice of number of doors and the (in my view) totally bogus part about Monty opening one of the other doors.

In fact, some people (in this very thread!) are still analyzing the problem as if the opened door represents new, 'very valuable' information! It is truly just slight of hand to make it seem like you're only picking one door when you are picking two doors.

[spronkey]

My theory on why it's such a difficult problem to understand is that it's too easy to get caught up in the real world problem, when this isn't actually a real world problem.

In the "real world" problem there are additional variables to consider, such as whether or not the host would open a door containing a car, whether that would result in a win or a loss, whether the host would offer a switch in all cases, whether if so, there is any additional incentive for switching (which wouldn't adjust the probability of winning, so is a red herring, but "intuitively" it seems that if the host is trying to make you lose by trying to make you switch, probability should go down).

When you try to conflate the real world problem, which actually has multiple different probabilities depending on exact scenario, into an answer to what's generally accepted as the "Monty Hall Problem", some of this gets in the way.

In addition, there are two probabilities in the "Monty Hall Problem" - the first is whether your switch results in a win. The second is the probability that, if you switch, it was the correct choice.

[PalmerEldritch]

Well except that you don't actually get to pick two doors. it wouldn't be much of a problem if it was posed as "You can pick 1 door or 2 doors, what do you want to do?" Since you only get to pick one door then Monty opening a door to reveal a goat does provide valuable information.