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timtyler comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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Re: a 50% chance you'd be on the heads path, and a 50% chance you'd be on the tails path.
Those are not the probabilities in advance of the experiment being perfomed. Once the experimental procedure is known the subjective probabilites for Beauty on awakening are 33% for heads and 67% for tails. These probabilities do not change during the experiment - since Beauty learns nothing.
"Once the experimental procedure is known the subjective probabilites for Beauty on awakening are 33% for heads and 67% for tails."
Suppose 50% of the population has some asymptomatic form of cancer. We randomly select someone and do a diagnostic test. If they have cancer (we don't tell them), we wake them up 9 times and ask their credence for cancer (administering amnesia-inducing drug each time). If they don't have cancer, we wake them up once.
The person selected for this experiment knows there is a 50% chance they have cancer. And they decide ahead of time that, upon awakening, they'll be 90% sure they have cancer. And this makes sense to you.