Monty Hall and intuition; a problem of phrasing?

Why is the solution to the Monty Hall problem counter to our intuition? Or is it? Are there people for whom the solution is intuitively obvious? If you belong in the latter category, I’d love to hear you out.

My guess is that the phrasing or presentation of the problem is problematic. I am convinced that if you re-present the problem in a different way, the solution may become more obvious.

And here’s a first attempt at re-phrasing. I’d love to get some feedback on whether or not this is helpful. Here goes:

The original phrasing (by Vos Savant):

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?

My attempt:

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. The host offers you to select two out of three doors. From these two, he’ll open one which shows a goat. Which door do you think will show the car? The door the host didn’t open, or the door that was not included in the initial selection?

What I think makes the solution more obvious is that the latter presentation guides one towards the fact that the initial selection has two-thirds probability of containing the car. This probability does not change and it is consequently transferred onto the unopened door in the initial selection.

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