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timtyler comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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Even though you knew ahead of time that there was a 50% chance you'd be on the heads path, and a 50% chance you'd be on the tails path, you'd shift those around without probability law justification?
I also think you are not careful with your wording. What does p(heads) on Monday mean? Is it a joint or conditional probability? p(heads | monday) = p(tails | monday), yes, but Beauty can't condition on Monday since she doesn't know the day. If you are talking about joint probabilities, p(heads and monday) does not equal p(tails and monday).
Re: "but Beauty can't condition on Monday since she doesn't know the day."
She could make a bet. You do not have to know what day of the week it is in order to make a bet that it is Monday.