It's time for a probability puzzle! Now, with more Futurama references!
In Universe A, Leela flips two coins. Fry saw her flip them, but didn't see how they came up. He is morbidly curious, and asks, "Are they both heads?" After Leela refused to answer, Fry asked, "Is at least one coin heads?" Leela reluctantly replied that at least one coin came up heads.
What is the probability that both coins are heads?
Meanwhile, in parallel Universe 1, Leela flips two coins. Fry saw her flip them, but didn't see how they came up. He politely asks, "How did they come up?" Leela, not wanting to give it all away, says, "At least one coin is tails."
What is the probability that both coins are tails?
Bonus question: A Tale of Interest
What if you are an external observer who has one additional piece of information to what I've previously shown? That is, you happen to know that Universe 1 and Universe A are intimately connected. Whenever a coin is flipped in Universe 1, it comes up on the opposite side in Universe A.
Given this new information, what is the probability that both coins are heads in Universe A?
Update: The solution has been posted.