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timtyler comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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We can tweak the experiment a bit to clarify this. Suppose the coin is flipped before she goes to sleep, but the result is hidden. If she's interviewed immediately, she has no reason to answer other than 1/2 - at this point it's just "flip a fair coin and estimate P(heads)". What information does she get the next time she's asked that would cause her to update her estimate? She's woken up, yes, but she already knew that would happen before going under and still answered 1/2. With no new information she should still guess 1/2 when woken up.
She knows in advance how many times she will be woken up (on each coin result). It says so in the problem description. So, she never answers 1/2 in the first place. She doesn't update on awakening. She updates when she is told the experimental procedure in the first place.
So, in the extreme sleeping beauty problem, when she is told the experimental procedure, she decides that it will be tails with near certainty?