The Sleeping Beauty problem, also known as the Sleeping Beauty paradox, is a puzzle in decision theory in which an ideally rational epistemic agent is told she will be awoken from sleep either once or twice according to the toss of a coin. Each time she will have no memory of whether she has been awoken before, and is asked what her degree of belief that “the outcome of the coin toss is Heads” ought to be when she is first awakened.

The problem was originally formulated in unpublished work in the mid-1980s by Arnold Zuboff (the work was later published as "One Self: The Logic of Experience") followed by a paper by Adam Elga. A formal analysis of the problem of belief formation in decision problems with imperfect recall was provided first by Michele Piccione and Ariel Rubinstein in their paper: "On the Interpretation of Decision Problems with Imperfect Recall" where the "paradox of the absent minded driver" was first introduced and the Sleeping Beauty problem discussed as Example 5. The name "Sleeping Beauty" was given to the problem by Robert Stalnaker and was first used in extensive discussion in the Usenet newsgroup rec.puzzles in 1999. A more recent paper by Peter Winkler discussing different sides of the problem was published in The American Mathematical Monthly in 2017.

As originally published by Elga, the problem was:

Some researchers are going to put you to sleep. During the two days that your sleep will last, they will briefly wake you up either once or twice, depending on the toss of a fair coin (Heads: once; Tails: twice). After each waking, they will put you back to sleep with a drug that makes you forget that waking. When you are first awakened, to what degree ought you believe that the outcome of the coin toss is Heads?

There are three superficial differences between Zuboff's unpublished versions, and the one Elga actually solved (which is not quite the same as the one he asked). They should not affect the solution method. Zuboff used a large number, N, of days. There was to be one waking per day after an unspecified coin-flip result, and one waking on a random day in that interval after the other result. Elga fixed N at 2, named Tails as the result where there were to be two wakings, and placed the one waking after Heads on day 1.

This has become the canonical form of the problem:

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

In either case, she will be awakened on Wednesday without interview and the experiment ends.