The fact that no new information was gained does not mean that your answer to "what do you think the coin flip was?" will neccessarily be equal to "what do you think the result of the coin flip will be?"
All it means is that you can already, before going to sleep, make an unconditonal strategy "when woken up, I'll say this". And your knowledge can easily lead you to a statement "right now, I believe the coin flip will be 50/50, but when I wake up I'll bet on tails".
The coin flip does change the probabilities significantly after you get awakened - e.g. in the 0:1 chance you know that being awakened implies that it must be that particular result, and the 1:100000000 scenario implies that if you're awakened then it either was tails or something extremely unlikely happened. The "no new information" clause simply means that you get already informed about this probability before going to sleep, not after the coin flip.
All it means is that you can already, before going to sleep, make an unconditonal strategy "when woken up, I'll say this". And your knowledge can easily lead you to a statement "right now, I believe the coin flip will be 50/50, but when I wake up I'll bet on tails".
The coin flip does change the probabilities significantly after you get awakened - e.g. in the 0:1 chance you know that being awakened implies that it must be that particular result, and the 1:100000000 scenario implies that if you're awakened then it either was tails or something extremely unlikely happened. The "no new information" clause simply means that you get already informed about this probability before going to sleep, not after the coin flip.