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[the other one is heads]

Sorry to resurrect an old thread, but I just found this, and there is a critical point missing.

So, if we real-timed this..

I happen to have a coin.  Before I flip it, what's the probability it will turn up heads twice?  25%.

I flip the coin once, and don't tell you what it was.  Before I flip the second one, what's the probability it will turn up heads twice?  25%.

I flip the coin a second time, and don't tell you what either coin is. What's the probability it will turn up heads twice?  25%.

I turn over one of the coins.  It's heads.  What's the probability it will turn up heads twice?


...oh.


OH.

I turn over one of the coins.  It's heads.  What's the probability the other one is heads?  50%



And if I ask you what were the chances both of them would have come up heads... would you say 25%?

Let's try it a different way:

• I happen to have two coins.  Before I flip either, what's the probability they will turn up the same way?  50%.
• I flip one coin and keep it hidden under a cup.  Before I flip the second one, what's the probability it will turn up the same way as they first?  50%.
• I flip the second coin, keeping it hidden as well. What's the probability they turned up the same way?  50%.
• I let you pick a cup, and I reveal its coin.  It's heads.  What's the probability the other is heads?

This is also 50%. If that isn't obvious from the progression, consider what you would have said if it had come up tails. The answer can't be different, so call it X for both questions. And if it is X for both questions, it is X if I just ask you to pick a cup and don't reveal its coin. And that's the same as the previous question.

The temptation to give a different answer is a paradox; and it even has a name. It is known as Bertrand's Box Paradox. The need to get the same answer, when the question is about symmetric options like  heads or tails, forces the answer to be the same as when nothing was revealed.

• Suppose, instead if you picking a cup, you ask me if either is heads. I look at both, and reveal one, showing that it is heads.  What's the probability the other came up the same way?

If both had been tails, originally a 25% chance, I couldn't have done this. Of the remaining 75%, 25% has two heads and 50% has a heads and a tails. So the answer is 25%/75%, or 33%.

Bertrand's Box Paradox doesn't apply, because the possible answers to your question aren't symmetric. The probability of two of the same result must be 100% if I can't reveal a heads.

• Suppose, instead of letting you pick anything, I look under both cups and show that one is a heads. What's the probability the other is heads?

This can't be answered without more information. If I decided to show a random coin (say, the left cup), the answer is 50% just like when you picked a cup. If I decided to show a heads if I could, the answer is 33%. And if I decided to show a tails if I could, but found I couldn't so I showed you a heads, the answer is 100%.

So the question is ambiguous. But if you had to pick one of those answers as "best," which would you pick? Can you apply Bertrand's Box Paradox?

The Florida question is essentially the same. The question is ambiguous, but maybe a "best" answer is possible. And it isn't the one usually given.