Sleeping beauty paradox

The Sleeping Beauty paradox is a paradox in probability created by Adam Elga.

The paradox imagines that 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 to determine which experimental procedure is undertaken. If the coin came up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin came 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 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?"

How should she answer? On the one hand, the coin is fair, so it seems that the answer must be 1/2. Yet each time she is awakened, she has to consider three possibilities—that the coin landed on heads and it is Monday, that the coin landed on tails and it is Monday, or that the coin landed on tails and it is Tuesday so the answer must be 1/3. This paradox highlights the importance of clearly formulated questions, as 1/2 and 1/3 are both correct answers, but to different questions.

Assume for the moment that when asked "What is your credence", Sleeping Beauty will try to give an objective assessment of the probability that the coin landed on heads. Much of the disagreement over this paradox stems from the common mistake of thinking that if an event has n possible outcomes, then the probability of each must be 1/n. This is often correct (for example, there is a 1/6 chance of a six-sided die landing on '2'), but not always so. A logical analogy of the Sleeping Beauty paradox is the following: suppose I toss a coin, and if it lands on heads I toss another coin; what is the probability that the first coin landed on heads? Clearly there are three possible outcomes (tails/tails, heads/tails, and heads/heads), but the probabilities are not equal. Since the second coin toss has no effect on the outcome of the first, the probability that the first toss results in heads is still ½. The second coin toss in this scenario is analogous to the question of whether it is Monday or Tuesday in our experiment.

However there is one subtly different version of the problem that yields another result. Suppose that Sleeping Beauty is not in fact trying to give an objective assessment of the probability, but is asked to guess whether the coin landed heads or tails, and is rewarded for each correct answer. Now she still has no idea whether the coin landed on heads or tails—there is a 50% chance of either—but the benefit is greater if the coin lands on tails, so she should guess tails. If she always guesses heads, then if she is right she will be right once, on the Monday. If she always guesses tails, then if she is right she will be right twice, on the Monday and the Tuesday. Always guessing tails will give her twice the reward on average, which is where the 1/3; figure comes from.

External links

• Usenet posting by Jamie Dreier starting the discussion (http://groups.google.com/groups?oi=djq&selm=an_455334937)
• Overview: Some "Sleeping Beauty" postings (http://www.maproom.co.uk/sb.html)
• "Sleeping Beauty" puzzle and solution (http://www.greylabyrinth.com/puzzles/puzzle.php?puzzle_id=puzzle075)
• Terry Horgan: Sleeping Beauty Awakened: New Odds at the Dawn of the New Day (http://dingo.sbs.arizona.edu/~thorgan/papers/Beauty.htm) (review paper with references)