I'd phrase it as a difference between decision theory and statistics. In almost all cases we're formulating advice, not predictions. In the monty-hall case, we

I'd phrase it as a difference between decision theory and statistics. In almost all cases we're formulating advice, not predictions. In the monty-hall case, we're rarely presented with a person who is going to help us in those circumstances.. so we are motivated to reason to an understanding of the situation in which that help is absent. ie., a better model of the scenario is harder to arrive at, because it involves making ecologically irrational assumptions on behalf of the motivations of others in a context (ie., game show) where suspicion is generally required. It's a perennial annoyance of mine that the weirdest people with the least fit social instincts (relevant academics) go around inveighing on supposed 'ordinary people' prescribing, most often, patterns of thinking that would get everyone killed. What the practice of formulating problems shows is that mathematicians (et al.) are extremely bad at phrasing problems to be intuitive. Indeed, I think they deliberately do the opposite, wishing to 'expose' the idiocy of others and their own superiority. Physicists do not practice physics this way. It would be insane to deliver a lecture where clear physical intuitions can be built, by instead choosing cases where they break down and saying, "ah! so you need the mathematics!". Physicists would, certainly, hold such an activity in contempt. Here we should be formulating these problems to be maximally intuitive, holding their structure constant. Then showing how the formalism helps.