Not sure what the answer is, ideas welcome!
You have 3 coins, one always comes up heads, the second always comes up tails, and the third is a fair coin. You select a coin at random. After selecting a coin, you flip it twice and get heads both times. What is the probability that your next flip is heads?
Follow up: Suppose you flipped twice and got the same outcome both times (either HH or TT), then what's the probability that your next two flips match the first two (e.g. if you saw HH, your next two are HH)