Judea Pearl & Dana Mackenzie, in their new book 
[*The Book of Why*](https://www.penguin.co.uk/books/289/289825/the-book-of-why/9780141982410.html) 
(p.190ff), explain the paradox in a way I hadn't seen before.

Pearl imagines changing the rules to "Let's Fake a Deal,"
where "Monty opens one of the two doors you didn't choose, but his choice is completely at random." Of course he could open the door
containing the car/prize, ruining the game. But his point is that, with this random choice, there is no longer any advantage to switching. If Monty opens
a door containing a goat, there is a $1$-in-$2$ probability of
winning either by sticking or by switching.

That Monty's door-opening is not random, but rather
constrained by Monty's need to avoid revealing the car, now makes it more clear that significant information is conveyed by his goat door-opening.