Problem #26 HARD
The Waking Uncertainty
Meta Microsoft Probability Philosophy
Problem Statement
A volunteer named Mira agrees to a probability experiment. She is put to sleep. A coin is flipped. If it lands heads, she is woken once on Monday. If it lands tails, she is woken on Monday and again on Tuesday — but between the two wakings her memory of the first waking is erased so she cannot tell which day it is. Every time she wakes, she is asked: 'What probability do you assign to the coin having landed heads?' What should she say?
Answer & Quick Explanation
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Most accepted answer: 1/3. Across repetitions, heads-wakings are one-third of all wakings. A 1/3 credence also maximizes decision-theoretic outcomes. The 1/2 answer is also coherent from a different philosophical frame — genuine disagreement exists among experts.
Detailed Editorial Solution
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This problem exposes a deep ambiguity in probability: should you reason over coin outcomes (halfer) or over observer-moments — i.e., the occasions on which you might be asked the question (thirder)?
Step 1: Halfer argument: The coin is fair. Mira was told she would be woken regardless of the outcome. Waking up is not news. P(heads) remains 1/2.
Step 2: Thirder argument by frequency: Run the experiment 1,000 times. Heads occurs ~500 times → 500 Monday wakings. Tails occurs ~500 times → 500 Monday + 500 Tuesday wakings = 1,000. Total wakings ≈ 1,500. Proportion that are heads-wakings: 500/1,500 = 1/3.
Step 3: Thirder argument by reference class: When Mira is awake, she knows she is in one of three equally probable scenarios: Heads-Monday, Tails-Monday, or Tails-Tuesday. Heads corresponds to exactly one of them.
Step 4: Halfer counter: The reference class of 'wakings' is not the same as the reference class of 'coin outcomes.' Conflating them is the error.
Step 5: Decision-theory test: If Mira bets $1 on heads each time she wakes, using probability 1/3 maximizes her long-run earnings. Using 1/2 does not. This supports the thirder answer in practical contexts.
Step 6: The halfer answer is defensible in a context where Mira is reasoning about the coin flip as a one-time event, not across repeated instances.
Key Insight:
The paradox reveals that probability is not purely about physical events — it also depends on how you define your reference class of uncertainty. When the number of times you might be asked depends on the outcome, the two frames diverge.