Thank you for walking me through this; I'm having a very hard time seeing the other perspective here.
I understand that P(Monday) is ambiguous. I meant to refer to "the probability that the current day, as Beauty is currently being interviewed, is Monday". Regardless of Beauty's perspective, she can ask weather the current day is Monday or Tuesday, and she does know that it is not currently both. And she can ask what the probability that the coin landed tails given that the current day is Tuesday, etc. Yes? Given that, I'm not seeing what part of my reasoning doesn't work if you replace each instance of "Monday" with "IsCurrentlyMonday".
I meant to refer to "the probability that the current day, as Beauty is currently being interviewed, is Monday".
What you just described is a per-awakening probability. Per-awakening, P(Heads) = 1/3, so the proof that P(Heads | Monday) > 1/2 actually only proves that P(Heads | Monday) > 1/3, which is true since 1/2 > 1/3.
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