Stunned by Sleeping Beauty: How Prince Probability updates his forecast upon their fateful encounter
Laurens Walleghem
TL;DR
This paper defends the view that Sleeping Beauty should assign $P(H)=1/3$ when awakened, by a straightforward Bayesian treatment that treats awakening as nontrivial information. It introduces an outsider, Prince Probability, who updates to $P(H)=1/3$ upon meeting an awake Beauty, while prior to contact he retains $P(H)=1/2$, illustrating how self-locating information governs updating. The authors extend the analysis to multiple awakenings, dreaming scenarios, copy-SB variants, and unfair coin tosses, showing that the $1/3$ conclusion persists across these settings and aligns with Elga’s Principle of Indifference-based reasoning. They also discuss connections to the Frauchiger–Renner paradox, arguing that the probabilistic extension challenges a core consistency assumption and supports a robust Bayesian account of self-locating belief. Overall, the work strengthens the case for $1/3$ as the correct updating rule upon awakening and clarifies how observer perspective shapes credence in Heads in the Sleeping Beauty problem and its variants.
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).