100% correct. The difference is, when one is already given the options and one is not, the one that is not often comes up with a broader range of solutions. It's proven over and over again. So you are correct. The wizard can think of the same solutions. But the real life results in psychology often state they don't. And that is because they are given a block of text that explicitly tells them what they are to do and its intended outcomes.
In other words, it's a game of chance, and human minds like a sure bet. So the wizard will err on the side of the sure bet and not even think about the optional "chance" categories.