[lnl]

> When you phrase it in a way that suggests the host's choice is itself random, people get it wrong.

In other words, they still get it right; they get it right for the separate question that that phrasing implies.

If the host's choice is random, so that when you initially picked wrong it's equally probable that the host open the door with the car and then say "sorry, looks like you lost" (which is what I assumed when I first heard this problem, not being familiar with the show), then even if the host happened to open the door with a goat and give you a chance to switch, it doesn't matter if you take it or not. People are correct that, for that question, the probabilities are one in two for both of the remaining doors.