What I think is true is that people underestimate the role of the host (i.e. the rules according to which he behaves to).
Or from [0]: "The standard annunciation of the MH problem, does not make explicit what I am assuming here: namely, that Monty will always open a door which does not contain a prize."
"Virtually all of my critics understood the intended scenario. I personally read nearly three thousand letters (out of the many additional thousands that arrived) and found nearly every one insisting simply that because two options remained (or an equivalent error), the chances were even. Very few raised questions about ambiguity, and the letters actually published in the column were not among those few." [0]
The crux of the problem is the counter-intuitive nature of probabilities and information, possible choices and choices that could have been. It's a difficult problem, purported to have tripped up even Paul Erdős.
[0] https://en.wikipedia.org/wiki/Monty_Hall_problem#Other_host_...