[PaultheS]

[ QUOTE ]
(1) Two coins are flipped. If they are both tails, we reflip both. We continue until we get an outcome that contains at least one head. What is the probability that we end up with two heads? The answer is 1/3, and your teacher is correct.

(2) A dime and a quarter are flipped. I check the dime and tell you whether it's heads or tails. What is the probability that the quarter is the same? Here the answer is 1/2, you are correct.

[/ QUOTE ]

This is true. Re-reading the question, it does sound like it could really mean either of these scenarios. However, I assume that when the teacher gave the question (if he really is a very intelligent guy) he probably outlined it more like the first scenario than the original poster did.