for this question, (the way it's worded above) you must stay as you have a 100% chance of winning if you stay.
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however before you open it the game host arbitrarily picks another empty door (not the one with the prize behind)

if you pick a door, and the the host is able to decide at random what door to pick (i.e it doesn't matter what door they pick), then you must have gotten the correct door to start with. (otherwise they would have to make an informed choice and could not pick at random).
Even if you say that the host doesn't pick arbitrarily, (i.e the host knows when the car is and know that he'll open one of two doors (whichever you don't pick).
I've read the statistical proof, and i (like many others) disagree.
in the first question, the change of you picking the prize is 1/3.
now the host has opened a door, you have a new choice, that bears no ground on the original question.
you now have a 1/2 chance of winning, so it doesn't matter.
there is a simillar probability question.
you see a woman walking down the street with a young girl, it's her daughter, she says that she has two children and one is at home, what are the odds that the other is a girl.
with two kids the possibly combinations are
BB 1/2 * 1/2 = 1/4
BG 1/4
GB 1/4
GG 1/4
you know that one of the kids is a girl so that wipes out the first combination
you're now left with
BG = 1/3
GB = 1/3
GG = 1/3
the idea is that there is a 2/3 possibility that the other child is a boy (2/3rd possibility of boy compared to 1/3rd possibility).
but it ignores the fact that it's a mutually exclusive event. (and random)
I just flipped two coins, one of the coins was a head, what's the other most likely to be?
the answer is that it's a 50/50 chance, the events are mutually exclusive, two independent choices.
in the original question, odds of getting the door right the first time are 1 in three.
now the host reduced the amount of doors to 2, forget your original choice and pick again (which is what the question do you want to swap really means)
the odds are now 1/2 it makes no difference if you change...
it's a different independent choice, so you don't compound the probabilities from the original question.
by the time the host has taken away a door that they know to be incorrect, then you're basically picking one of two, with an equal chance of the prize being behind either door.
so yes,
if you take the view that the host is asking you if you want to stick with your original choice 1/3
or now that he's opened the other door you can effectively have both doors (2/3) then I can see why they say switch...
on the other hand, you know that one of those doors does not contain the prize, you're not seeing 2 doors for the price of 1 by switching, one of them is empty, the odds aren't 2/3 one is definitely not the answer it's still 1/3 (you've seen the third door. if you stay you don't unsee it, if you stay you get to see what's behind your door and the open door, so you get 2/3 odds the same as if you switch). with 1/3 odds on the unpicked unseen door.
but now we've taken the third door away it's 1/2 staying, or 1/2 switching.