You are right, but your analogy doesn’t fit the Monty Hall question.
The outcome you are betting on (is it heads or tails, behind what door is the prize located) doesn’t actually change, the winning door doesn’t change, only the amount of doors you have to choose between changes.
So a better analogy, that may make it more intuitive, goes like this:
Someone tosses a coin but doesn’t show you what side landed up, you get to say what side you think landed up, the person that tossed the coin tells you what side didn’t land up, you get to change your answer based on this new information.
Either the parents went to the headmistress with Robin or they didn’t (50% either way). If they did, then it’s 100% chance that Robin is a girl. If they didn’t, there’s 50% chance that Robin is a girl, because you have no information about Robin.
I think it’s perfectly reasonable to come up with a situation where “You learn that they have two children, one of which is a girl” still gives no other information about the other.
Similar to @Richard_Kirk’s example above (but without names): you hear they have two children, and you see them walking down the street with one of their children who is a girl. I think in this matches the intent of the puzzle, so it’s 33% chance that the other is a girl.
Not related to your comment, but for the puzzle in general: One thing that I didn’t take account of before thinking about it: I originally thought the 33% number seemed surprisingly low. The I realized that the odds had actually increased from 25% chance that they had two girls to 33% chance, given you know one is a girl. That made everything seem much more intuitive.
