Sleeping Beauty problem

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The sleeping beauty problem is a puzzle in probability theory: a sleeper is to be woken once or twice according to the toss of a coin, and asked her credence for the coin having come up heads.

The problem was originally stated by Adam Elga^[1] but is based on earlier problems of imperfect recall^[2] and the older "paradox of the absentminded driver".

1 The problem
2 Solutions
3 Variations
4 Notes
5 References
6 See also
7 Other works discussing the Sleeping Beauty problem
8 External links

[edit] The problem

The paradox imagines that 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 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 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?"

[edit] Solutions

This problem is considered paradoxical because the answer can be seen as either 1/2 or 1/3.

On the one hand, the coin is fair, so it seems at first sight that the answer must be 1/2. Yet each time she is awakened, she actually 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. Since only one of the three options is the result of a heads, 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.

The reason there are two solutions is that one solution tries to minimize expected total inaccuracy, while the other tries to minimize expected average inaccuracy. Is it based on the percentage of runs of the experiment where the coin comes up tails? Or is it based on the percentage of interrogations where the coin comes up tails?

[edit] Variations

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.

[edit] Notes

The days of the week are irrelevant, but are included because they are used in some expositions.

[edit] References

^ Self-locating belief and the Sleeping Beauty problem by Adam Elga
^ "Sleeping Beauty" postings

[edit] See also

Credence — the subjective estimate of probability.

[edit] Other works discussing the Sleeping Beauty problem

Arntzenius, F. (2002) Reflections on Sleeping Beauty, Analysis, 62-1, 53-62
Bostrom, Nick (2002-07-12). Anthropic Bias. Routledge (UK), 195-96. ISBN 0-415-93858-9.
Bruce, Colin (2004-12-21). Schrodinger's Rabbits: Entering the Many Worlds of Quantum. Joseph Henry Press, 193-96. ISBN 0-309-09051-2.
Bradley, D. (2003) Sleeping Beauty: a note on Dorr's argument for 1/3, Analysis, 63, 266-268
Dorr, C. (2002) Sleeping Beauty: in Defence of Elga, Analysis, 62, 292-296
Elga, A. (2000) Self-locating Belief and the Sleeping Beauty Problem, Analysis, 60, 143-147
Lewis, D. (2001) Sleeping Beauty: Reply to Elga, Analysis, 61, 171-176
Monton, B. (2002) Sleeping Beauty and the Forgetful Bayesian, Analysis, 62, 47-53