For 1-9, tails being the correct guess 90% of the time still makes sense to me.
If you run the experiment twice, typically the first time you get heads, you are woken once, and so heads is correct answer once, the second time you get tails, but you are woken 9 times, so tails is the correct answer 9 times.
Even from the frame of reference of the experimenter, heads and tails are equally likely, but it if you guess heads incorrectly, you will be wrong 9 times, or 9 times as wrong ;).
Basically it depends on what you consider to be a "trial" of the experiment, if a coin toss is one trial, the wakings are a red herring, the probability is the same as the coin toss. If one trial is one waking, 90% of wakings are going to be tails.
The phrasing of the question is When you are awakened, to what degree should you believe that the outcome of the coin toss was Heads?
It's not actually asking for a strategy about guessing the correct answer, or how many times you'll be right or wrong.
It think whether you are a halfer or a thirder is going to depend on how you interpret the question.
If you see it as asking about the frequency at which "Heads" will the right answer, then it's clearly 1/3. But that's not what the question actually asks (probably - all language is interpreted).
If you run the experiment twice, typically the first time you get heads, you are woken once, and so heads is correct answer once, the second time you get tails, but you are woken 9 times, so tails is the correct answer 9 times.
Even from the frame of reference of the experimenter, heads and tails are equally likely, but it if you guess heads incorrectly, you will be wrong 9 times, or 9 times as wrong ;).
Basically it depends on what you consider to be a "trial" of the experiment, if a coin toss is one trial, the wakings are a red herring, the probability is the same as the coin toss. If one trial is one waking, 90% of wakings are going to be tails.