What information, exactly, does she get in the middle of the experiment that she cannot anticipate beforehand?
That she's the one in the experiment. She can't anticipate it before-hand because she doesn't know the probability of being in the experiment. It depends on whether the coin lands on heads or tails.
Imaging someone takes a deck of cards. They then flip a coin. On heads, they add a joker. On tails, they add two. They don't show you the result. You then draw a card. Can you anticipate the probability of getting a joker? If the only observers in the universe were created solely for that experiment, and each of them was given one of the cards, would that change anything?
She can still anticipate the possibility of being in the experiment and therefore make a strategy for what to do if she turns out to be in the experiment. That is all I'm doing here.
If she comes up with a strategy for what to do if she wakes up in the experiment, and then wakes up in the experiment, she doesn't get any additional information that would change her strategy.
leeping Beauty is put to sleep on Sunday. If the coin lands on heads, she is awakened only on Monday. If it lands on tails, she is awaken on Monday and Tuesday, and has her memory erased between them. Each time she is awoken, she is asked how likely it is the coin landed on tails.
According to the one theory, she would figure it's twice as likely to be her if the coin landed on tails, so it's now twice as likely to be tales. According to another, she would figure that the world she's in isn't eliminated by heads or tails, so it's equally likely. I'd like to use the second possibility, and add a simple modification:
The coin is tossed a second time. She's shown the result of this toss on Monday, and the opposite on Tuesday (if she's awake for it). She wakes up, and believes that there are four equally probable results: HH, HT, TH, and TT. She then is shown heads. This will happen at some point unless the coin has the result HT. In that case, she is only woken once, and is shown tails. She now spreads the probability between the remaining three outcomes: HH, TH, and TT. She is asked how likely it is that the coin landed on heads. She gives 1/3. Thanks to this modification, she got the same answer as if she had used SIA.
Now suppose that, instead of being told the result of second coin toss, she had some other observation. Perhaps she observed how tired she was when she woke up, or how long it took to open her eyes, or something else. In any case, if it's an unlikely observation, it probably won't happen twice, so she's about twice as likely to make it if she wakes up twice.
Edit: SIA and SSA don't seem to be what I thought they were. In both cases, you get approximately 1/3. As far as I can figure, the reason Wikipedia states that you get 1/2 with SIA is that it uses sleeping beauty during the course of this experiment as the entire reference class (rather than all existent observers). I've seen someone use this logic before (they only updated on the existence of such an observer). Does anyone know what it's called?