Stunned by Sleeping Beauty: How Prince Probability updates his forecast upon their fateful encounter

TL;DR

This paper clarifies the Sleeping Beauty problem using Bayesian probability, arguing that the correct probability for Heads upon awakening is 1/3, and explores how additional observers and knowledge affect this belief.

Contribution

It provides a Bayesian analysis confirming 1/3 as the correct probability and introduces the concept of an observer, Prince Probability, to strengthen the argument.

Findings

Bayesian reasoning supports the 1/3 probability for Heads.

Additional observer interactions influence belief updates.

The analysis extends to Sleeping Beauty's knowledge of dreaming.

Abstract

The Sleeping Beauty problem is a puzzle in probability theory that has gained much attention since Elga's discussion of it [Elga, Adam, Analysis 60 (2), p.143-147 (2000)]. Sleeping Beauty is put asleep, and a coin is tossed. If the outcome of the coin toss is Tails, Sleeping Beauty is woken up on Monday, put asleep again and woken up again on Tuesday (with no recollection of having woken up on Monday). If the outcome is Heads, Sleeping Beauty is woken up on Monday only. Each time Sleeping Beauty is woken up, she is asked what her belief is that the outcome was Heads. What should Sleeping Beauty reply? In literature arguments have been given for both 1/3 and 1/2 as the correct answer. In this short note we argue using simple Bayesian probability theory why 1/3 is the right answer, and not 1/2. Briefly, when Sleeping Beauty awakens, her being awake is nontrivial extra information that leads her to update her beliefs about Heads to 1/3. We strengthen our claim by considering an additional observer, Prince Probability, who may or may not meet Sleeping Beauty. If he meets Sleeping Beauty while she is awake, he lowers his credence in Heads to 1/3. We also briefly consider the credence in Heads of a Sleeping Beauty who knows that she is dreaming (and thus asleep).