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The Sleeping Beauty Problem

I’m not sure whether this is totally a math problem or a logic problem, but it has elements of both.

We plan to put Beauty to sleep by chemical means, and then we'll flip a
(fair) coin. If the coin lands Heads, we will awaken Beauty on Monday
afternoon and interview her. If it lands Tails, we will awaken her Monday
afternoon, interview her, put her back to sleep, and then awaken her again
on Tuesday afternoon and interview her again.

The (each?) interview is to consist of the one question: what is your
credence now for the proposition that our coin landed Heads?

When awakened (and during the interview) Beauty will not be able to tell
which day it is, nor will she remember whether she has been awakened
before.

She knows the above details of our experiment.

What credence should she state in answer to our question?

Now, Sleeping Beauty is the lamest of all the Disney Princesses, and never struck me as smart enough to even know what credence means. That’s beside the point, because the subject could be anyone.

Most of us would have a 50% belief that the coin landed on heads, without taking the rest of the experiment into consideration, because it’s a coin toss. I would think that not remembering a past awakening would make the day of the week moot to the person being awakened.

But there are those who crunch the numbers differently, and say the chance of the coin toss landing on heads would only be 33%. See, there are three possible outcomes.

Heads/Monday
Tails/Monday
Tails/Tuesday

And the probability of any of them is one third. But the way that “thirder position” is explained at Wikipedia would lead me to wake up and say there’s a good chance it’s Monday and you math nerds haven’t even tossed a coin at all yet.

What do you think?

The possible answers and their reasonings are still being argued about. There are discussions at both Google and Metafilter with people giving various explanations for their answers.


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