I would have thought that most times someone starts going down that road - of thinking that the host's choice had been a random selection from the two remaining doors - she would immediately see that it would be the wrong way to think about the problem because the prize would sometimes be revealed, a situation that does not occur in the question as posed, and which, when it does occur, renders the question of switching moot.
In other words, if one starts by assuming that the question implies a random choice, it quickly becomes apparent that the question would not be well-formed under that assumption. To be a well-formed question under that assumption, it would have to say something like "the host flips a coin to pick one of the closed boxes which, when opened, reveals a goat", and that is a different question, not the original one rephrased.
In other words, if one starts by assuming that the question implies a random choice, it quickly becomes apparent that the question would not be well-formed under that assumption. To be a well-formed question under that assumption, it would have to say something like "the host flips a coin to pick one of the closed boxes which, when opened, reveals a goat", and that is a different question, not the original one rephrased.