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Judea Pearl & Dana Mackenzie, in their new book The Book of Why (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.

He assume you open Door $1$, and then:

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (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.

He assumes you open Door $1$, and then:

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (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.

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (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.

He assume you open Door $1$, and then:

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (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 his need to avoid revealing the car, now makes it more clear that significant information is conveyed by his door-opening.

Judea Pearl & Dana Mackenzie, in their new book The Book of Why (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.