The probability is 1/3 per awakening and 1/2 per experiment.
Per-experiment:
Per-awakening:
I don't see anything in either of those links claiming that P(Heads | Monday) > 1/2. I assume that your reasoning to get that is something like "P(Heads | Tuesday) is less than P(Heads), so it follows that P(Heads | Monday) is greater than P(Heads)." However, if you're calculating per-experiment, Monday and Tuesday are not mutually exclusive, so this reasoning doesn't work. (If you're calculating per-awakening, P(Heads) isn't 1/2 anyway.)
Some additional support for the apparently unreasonable conclusion that if it is Monday, it is more likely that the coin will land heads. Suppose that on each awakening, the experimenter flips a second coin, and if the second coin lands heads, the experimenter tells Beauty what day it is, and does not do so if it is tails.
If Beauty is told that it is Tuesday, this is evidence (conclusive in fact) that the first coin landed tails. So conservation of expected evidence means that if she is told that it is Monday, she should treat this as evidence that the first coin will land heads.
If it's worth saying, but not worth its own post, then it goes here.
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