Problem Setup

• Sleeping Beauty agrees to experiment, knows all details beforehand
• Put to sleep Sunday, fair coin flipped
• If Tails: Wake up Monday → ask questions → amnesia drug (no memory) → wake up Tuesday → ask questions → amnesia drug
• If Heads: Wake up Monday only → ask questions → amnesia drug
• Experiment ends Wednesday, Beauty freed
• Key question asked each awakening: "What's probability coin was heads?"

Halfer Argument (1/2 probability)

• Fair coin flipped Sunday = 50/50 chance
• Sleeping Beauty learns no new information upon waking
• Cannot apply Bayes' theorem or conditional probability without new data
• Probability remains what it always was: 1/2
• Being woken up doesn't change original coin flip odds

Thirder Argument (1/3 probability)

• Beauty doesn't know which day she woke up
• Assume Tails scenario: Could be Monday or Tuesday equally likely
  • P(Monday|Tails) = P(Tuesday|Tails)
  • Therefore: P(Monday ∩ Tails) = P(Tuesday ∩ Tails)
• Assume Monday scenario: Could be Tails wake-up or Heads wake-up equally likely
  • P(Tails|Monday) = P(Heads|Monday)
  • Therefore: P(Tails ∩ Monday) = P(Heads ∩ Monday)
• Three equally likely scenarios:
  1. Monday + Tails
  2. Tuesday + Tails
  3. Monday + Heads
• Each has probability 1/3, only one involves Heads
• Therefore P(Heads) = 1/3

Rip Van Winkle Extension

• Modified experiment: Heads = 1 wake-up, Tails = 999 wake-ups
• 1,000 total possible wake-up scenarios
• Only 1 scenario involves Heads
• Intuitive answer: P(Heads) = 1/1,000
• Supports thirder position through extreme case

Philosophical Resolution Discussion

• Both mathematical arguments equally valid
• Paradox stems from question interpretation
• Original 2000 Elga formulation asked about "credence" (strength of belief)
• Key distinction:
  • Are we asking about coin flip probability?
  • Or asking which experimental path Beauty is on?
• Beauty more often correct saying "Tails" but gets more chances when Tails occurs
• Resolution depends on precise question interpretation

Betting Scenario Analysis

• If rewarded for correct answers: Beauty should always say Tails
• Gets 2 chances to be right with Tails vs 1 chance with Heads
• But this doesn't necessarily mean coin was more likely Tails
• Highlights difference between "path probability" vs "coin probability"