It's time for a probability puzzle! Now, with more Futurama references!

Question A:

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?

Question 1:

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.

## Tuesday, May 6, 2008

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## 4 comments:

So, I'm thinking that the probability of 2 heads is 50%, and the probability of 2 tails is also 50%.

I can't quite see why the additional information makes any difference, but I'm definitely not good at intuitive probability, I'd probably need to make a decision tree to work it out.

You're right... in one of the universes.

Hmmm…this is one of those evil counterintuitive problems like the Monty Hall problem, isn't it? I'd say that in Universe A, the probability that both coins are heads is 1/3. There are four possible ways the coins could have landed, three of which have at least one head. Assuming Leela is guaranteed to answer honestly, she is guaranteed to answer yes for all three possibilities (which have equal probability) and no for the tails-tails possiblity. Thus given that at least one is heads, the probability that both are heads is 1/3.

I don't see a logical difference for Universe 1, except that Leela's choice of how to answer may convey additional information. I definitely smell a whiff of Monty Hall, though.

Give me a bit more time to think about the bonus question.

I don't think there is any information in Leela's answer in Universe 1, the way there is in Universe A, because she looked at the coins before deciding to answer whether there was "at least one coin is tails". She could have just as easily decided to say that at least one coin is heads, or both coins are the same, because she didn't answer specifically Fry's question. No matter what the coins were, she could have said something.

So in Universe A, she answered Fry's question, "is at least one coin heads?", and by saying yes, she eliminated 25% of the possibilities. (The 3 remaining possibilities being 1)coin A is heads, coin B is tails; 2) coin A is tails, coin B is heads; and 3) both coins are heads.) So the probability that both are heads is 1/3.

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