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neq1 comments on Beauty quips, "I'd shut up and multiply!" - Less Wrong
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"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.