I love this puzzle, but it's better (IMHO) expressed as a card game.

You have three cards. One is white on both sides, one is black on both sides and one is white on one side and black on the other.

You choose a card at random and lay it on the table. It's white. What are the chances that the other side is also white?

Everyone says 1/2 (wrongly, as you point out). Then you say, "You take a card at random. What are the chances that both sides are the same?"

Everyone says 2/3, and likely the penny drops at this point, because seeing white or black really gives you no useful information.