This paper presents a novel approach to resolving the Sleeping Beauty problem by drawing an analogy with a hypothetical “hyper-entanglement urn.”
The Hyper-Entanglement Urn:
I begin by introducing a thought experiment involving an urn containing red and green balls. Some red balls are hyper-entangled with green balls, meaning that removing a red ball also removes its entangled green counterpart, and vice versa. This urn serves as an analogy for understanding the Sleeping Beauty problem, highlighting the unique properties of hyper-entangled objects.
The Sleeping Beauty Problem:
Sleeping Beauty is put to sleep and awakens without memory of previous awakenings. The probability of the coin toss (Heads or Tails) determines her awakenings. The problem is to determine the probability that the coin landed Heads upon her awakening. Two main perspectives exist: the ‘halfer’ view (probability is 1/2) and the ’thirder’ view (probability is 1/3). I argue that these perspectives are limited and proposes to consider the problem more broadly.
The Urn Analogy:
I suggest that the standard urn analogy used in both the halfer and thirder arguments is insufficient. I argue that the Sleeping Beauty problem is more accurately represented by the hyper-entanglement urn. This is because, in the case of Tails, the awakenings on Monday and Tuesday are inseparable, much like the hyper-entangled balls. This insight leads to a new way of conceptualizing the problem.
Consequences of the Analogy with the Hyper-Entanglement Urn:
The paper explores the implications of this new analogy. It suggests that considering the balls as either individual colors or as hyper-entangled objects (pairs) leads to different probabilities. This distinction mirrors the halfer and thirder perspectives but within a unified framework. I show that depending on whether we focus on the color or the object nature of the balls, we arrive at different probabilities for drawing a red (or green) ball, which parallels the probabilities of waking on Monday or Tuesday in the Sleeping Beauty problem.
A Two-Sided Account:
The paper synthesizes the insights from the halfer and thirder perspectives within the context of the hyper-entanglement urn. It argues that both views have merits but are incomplete when considered separately. By using the urn analogy, I demonstrate that the probability of drawing a red ball (or waking on Monday) can be understood in two ways, depending on whether we focus on the color or the object nature of the balls. This two-sided approach reconciles the conflicting views in the Sleeping Beauty problem.
Implications for Conditional Probabilities and the Sleeping Beauty Problem:
I extend the analysis to conditional probabilities and the overall probability of Heads upon awakening. I distinguish between probabilities calculated from the perspective of balls-as-color and balls-as-object, leading to different results for the probability of Heads. This nuanced approach offers a resolution to the Sleeping Beauty problem that acknowledges the validity of both halfer and thirder arguments while transcending their limitations.
Conclusion and Philosophical Implications:
The paper concludes by reflecting on the broader philosophical implications of this approach. I suggest that the solution to the Sleeping Beauty problem requires considering unconventional objects (like hyper-entangled pairs) in probabilistic reasoning. This insight aligns with Nelson Goodman’s philosophy, which emphasizes the importance of considering diverse kinds of objects in our understanding of the world.