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Comment by te_platt | Hacker News Reader

Your reframing of the question changes the problem. Let A be the probability of Heads (1/2) and B be the probability you are asked (1/2) then P(B|A) is 1/3. So the probability the coin came up heads is P(A|B) = P(B|A) * P(A)/P(B) = 1/3. Which is what you expected.

In the original problem the probability of getting woken up is 1. So we end up with the probability of the coin being heads given that I was just woken up is (1 * 1/2) /1 = 1/2.

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Suppose I am going to flip a coin and give you money if you guess correctly. On heads I hand you 1$ on tails I hand you 1$ and add 1$ to your bank account without telling you. Now, after flipping the coin I ask you what the heads or tails?

Clearly your going to pick tails. Why? Because it's the more rational choice based on the spoken rules.

Going to sleep and losing memory is really just window dressing. The simple truth is before you went to sleep you should be able to say 1/3 odds of waking up to heads.

But that's still not the question.

The stated question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?

Not, Should you _guess_ Heads or Tails?

The probability of the coin toss _does not change_. The probability that "the reason they are waking you up is because the coin was Heads" does change, but that's not the question.

Clearly there is a 50:50 shot you get heads or tails from the coin. In your model of it's a 50:50 coin then you would have 50:25:25. The problem is waking up the second time does not reduce the odds of waking up the first time thus 50:50:50.

I think you changed the question. Before you go to sleep you know you have a 50% chance of waking up to heads and a 50% chance of waking up to tails twice. When you wake up you haven't learned anything so those probabilities can't have changed.

You have arguably less information when you wake up in that you don't know the time. Are you waking up because it's tails? because it's heads? or is it tails the second time?

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