A probability puzzle that has philosophers fiercely divided. Is the answer 1/2 or 1/3? Both sides have compelling arguments.
Sleeping Beauty volunteers for an experiment. On Sunday, she's told the rules and put to sleep. A fair coin is flipped.
If Heads: She's woken on Monday, interviewed, and the experiment ends.
If Tails: She's woken on Monday, interviewed, put back to sleep with her memory erased, then woken on Tuesday and interviewed again.
Each time she wakes, she's asked: "What is your credence that the coin landed Heads?"
"What is your credence that the coin landed Heads?"
The coin is fair. Beauty gains no new information upon waking—she knew she would wake up regardless of the outcome. Her credence should match the coin's probability: 50%.
There are three possible awakening states, all indistinguishable to Beauty. Only one of three (Monday-Heads) involves Heads. By the principle of indifference: 33.3%.
No New Information: Beauty knew before the experiment that she would be awakened. Waking up provides no new evidence about the coin flip. Her prior probability (1/2) should remain unchanged.
David Lewis's Position: Credence should track objective probability. The coin is fair, so P(Heads) = 1/2, regardless of the awakening structure.
Self-Location Uncertainty: Beauty is uncertain about which of three possible awakening-states she's in. By the Principle of Indifference, she should assign equal probability to each: 1/3.
Betting Argument: If Beauty bets $1 that it's Heads each time she wakes, she loses $2 when Tails (two bets) but wins $1 when Heads (one bet). Fair odds require P(Heads) = 1/3.
Run the experiment many times. Count awakenings, not experiments. Across all awakenings, only 1/3 occur when the coin was Heads. This is what Beauty experiences—her "credence" should match this frequency.
But halfers counter: "Credence about the coin" differs from "credence about which awakening this is."
Some philosophers argue both answers are correct—to different questions! "What's the probability the coin landed heads?" (1/2) vs. "What's the probability this awakening is a Heads-awakening?" (1/3). The puzzle's ambiguity is the real lesson.
The Sleeping Beauty problem was first introduced by Adam Elga in his 2000 paper "Self-locating belief and the Sleeping Beauty problem," where he argued for the 1/3 position.
Philosopher David Lewis quickly responded with the halfer position, sparking an ongoing debate that continues today.
Though most philosophers favor the thirder position, there is no consensus on what the best argument for it is. The problem has become a touchstone for understanding probability, belief updating, and personal identity.