Let's assume you're the so-called peeker, Eddie. You look under the cup and see a 2 but do not look at the other die. You report "at least one of the dice is a 2". You also will answer your own question that the probability of there being 2-2 is 1 in 6--not 1 in 11, because you know which of the two dice shows a 2 at the moment.
Alan, OTOH, also "knows" it's 1 in 6 because in his mind he has taken one of the dice completely out of the picture, leaving only one die. But what does a WoV armchair "AP" have to say about this? Why....1 in 11 of course! And this is only because they are assuming nobody but the peeker knows what he has seen.
Day-in/day-out the prevailing answer of 1 in 6 will be correct, except in the flawed world of theorists, who will never admit that the original question was incompetently formulated. The OP outsmarted himself, because he never bounded the question with necessary absolutes. And for a forum loaded with supposed "mensas" how surprising it is to see one of them fail so extraordinarily.