The Sleeping Beauty Paradox is a question of how anthropics affects probabilities.
Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.
Each interview consists of one question, “What is your credence now for the proposition that our coin landed heads?”
One argument says that since Beauty will see the same thing on waking whether the coin came up heads or not, what she sees on waking provides no evidence one way or the other about the coin, and therefore she should stick with the prior probability of one half.
Another argument replies that the two awakenings when the coin comes up tails imply that waking up itself should be considered evidence in favor of tails. Out of all possible situations where Beauty is asked the question, only one out of three has the coin showing heads. Therefore, one third.
A third argument tries to add rigor by considering monetary payoffs. If Beauty’s bets about the coin get paid out once per experiment, she will do best by acting as if the probability is one half. If the bets get paid out once per awakening, acting as if the probability is one third has the best expected value.