I think the problem is in the phrasing of the question: When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?
That's not a precise probability question.
More significantly the question (whatever it might be) is potentially asked more than once.
So, if the question is "what is the probability that the coin-flip was Heads?" (which I think it the most likely interpretation) then it's 1/2
But if the question is "what is the probability that you are being woken up because the coin was Heads" then it's 1/3
Asking me what I "should believe about the outcome" is (intentionally, I suppose) blurring the question so that it disguises the fact that there are actually 2 different questions in play.
The event isn't just "there was a coin flip". The event was "there is a coin flip and I have been woken up". For this event, two thirds of the time the coin flip will have been tails.
Halfers forget that there is an extra piece of information available "I have been woken up". SB knows that one third of the time she has been woken up it is because the coin flip was heads, and two thirds of the time it was tails.
Anyway, this is a stupid "paradox". It is extremely simple to build a simulation of SB being woken up and coin tosses, and if you do so, you get the clear answer - 1/3. There is no sampling issue or other trick that makes this simulation hard to write correctly...
No, those questions are equivalent. In the single experiment case, the answer to both is 1/2, and in the infinite case, it's 1/3. There's no "3" to even enter into the equation in the single experiment case.
That's not a precise probability question.
More significantly the question (whatever it might be) is potentially asked more than once.
So, if the question is "what is the probability that the coin-flip was Heads?" (which I think it the most likely interpretation) then it's 1/2
But if the question is "what is the probability that you are being woken up because the coin was Heads" then it's 1/3
Asking me what I "should believe about the outcome" is (intentionally, I suppose) blurring the question so that it disguises the fact that there are actually 2 different questions in play.