Kewl you wrote "it's done before the roll." That alters reality.

The original problem specifically states that two dice were rolled and at least one is showing a 2. Whatever you thought prior to that no longer matters. Now you have to deal with at least one die showing a two. The question now centers on the second die in the problem which may or may not be a two and the chances for that second die to be a 2 is 1/6.

By figuring probability before the roll alters the reality of the question. You are changing the facts and the conditions of the problem.

As I have said a hundred times, your 1/11 is "good math" but for a different question -- perhaps for a question that asks: "how many different combinations of two dice have at least one 2 and of those combinations how many would show 2-2?" That would be your 1/11 answer.