I quite like your phrasing because it highlights that probability is a model of uncertainty for an observer. So the mother has no information on the door, so s

I quite like your phrasing because it highlights that probability is a model of uncertainty for an observer. So the mother has no information on the door, so she is 'neutral' (p=1/2 of guessing right); you have a small amount of information (p=2/3); you can even include Monty which of course has total information (p=1). You can't ask what are the probabilities of finding goats behind each door without specifying an observer and the available information.