Today is Sunday. Sleeping Beauty drinks a powerful sleep potion and falls asleep.
Her attendant tosses a fair coin and records the result.
In essence, the awakenings on Mondays and Tuesdays are indistinguishable to her.
The most important question she's asked in the interviews is
What is your credence (degree of belief) that the coin landed in heads?
Given that Sleeping Beauty is epistemologically rational and is aware of all the rules of the experiment on Sunday, what should be her answer?
There are three cases when Beauty is awakened. When it is (Tails and Monday), (Tails and Tuesday) and (Heads and Monday).
But since all of these situations are indistinguishable, she believes that the current situation is (Heads and Monday) with credence , by virtue of The Principle of Indifference. But since that is the only case where we have heads, she believes that the coin landed in heads with one-third credence.
Since Beauty knows all the rules of the experimental setup, her credence that the coin landed in heads is before she drinks the potion on Sunday.
Since she receives no new information when awaken, she continues to believe that the coin landed in heads with the same credence.
To quote wikipedia,
The Sleeping Beauty puzzle reduces to an easy and uncontroversial probability theory problem as soon as we agree on an objective procedure how to assess whether Beauty's subjective credence is correct.
However, this is not a problem on Probability theory but the concept of subjective probabilities or credences and whether they are well defined concepts.
I am slightly surprised to see that no one has posted this on Brilliant before.