Self-Locating Probability
The Sleeping Beauty Problem
A coin is flipped Sunday. If heads, you wake once. If tails, you wake twice (with memory erased). You wake up. What's P(heads)?
Here is the setup: Sleeping Beauty volunteers for an experiment. On Sunday, she is put to sleep. A fair coin is flipped.
If HEADS: Beauty is awakened on Monday, interviewed, and the experiment ends.
If TAILS: Beauty is awakened on Monday, interviewed, put back to sleep with her memory of the awakening erased, then awakened again on Tuesday and interviewed.
Beauty wakes up. She knows all the rules. She is asked:
"What is your credence that the coin landed heads?"
This seemingly simple puzzle has divided philosophers for over two decades. There are two main camps:
Halfers say 1/2
"The coin is fair. Nothing happened to change this. P(Heads) = 1/2."
Thirders say 1/3
"Two-thirds of awakenings are tails-awakenings. P(Heads) = 1/3."
Both positions are defended by serious philosophers. Both have compelling arguments. Let's explore why this problem is so hard.
The Experiment Protocol
First, let's make sure we understand exactly what happens. Watch the experiment unfold step by step:
Sunday
Coin Flip
Monday
Memory Wipe
Tuesday
Click "Run Experiment" to see the Sleeping Beauty protocol in action.
Run the experiment several times to see both heads and tails outcomes.
The Halfer vs. Thirder Debate
The disagreement comes down to how Beauty should update her beliefs when she wakes up. Click each position to see its full argument:
Halfer
1/2"The coin is fair. Nothing happens during the experiment to change this. P(Heads) = 1/2."
Thirder
1/3"There are three possible awakenings: Monday-Heads, Monday-Tails, Tuesday-Tails. Only one is a heads-awakening. P(Heads) = 1/3."
All possible awakening states:
Heads
Tails
Tails
Thirders: Each state equally likely (1/3 each). Halfers: H-Mon = 1/2, T-Mon = 1/4, T-Tue = 1/4.
The Thirder Argument: Count Awakenings
The Thirder argument is intuitive when you run the experiment many times. If you sample a random awakening, what fraction are heads-awakenings?
Run many experiments and count awakenings. Among all awakenings, what fraction occur in heads-experiments vs tails-experiments?
The simulation shows that roughly 1/3 of awakenings occur in heads-experiments, while 2/3occur in tails-experiments. Thirders argue that Beauty should reason as if she is randomly sampled from all possible awakenings.
The Betting Argument
Here's a compelling argument for Thirders: consider how Beauty should bet. If she bets $1 at even odds on each awakening, which strategy wins?
You are Beauty. Each awakening, bet on whether the coin landed heads or tails. You win 1:1 on correct bets. What strategy maximizes your winnings?
Your Balance
$100
Bet Amount
Total Bets
0
Even Halfers often agree that Beauty should bet as if P(Heads) = 1/3. But some argue this shows credence and betting can come apart - you can believe P(Heads) = 1/2 while betting as if it were 1/3.
The Halfer Defense
Halfers have a powerful argument: Beauty gains no new information when she wakes up. She knew in advance that she would wake up at least once. Why should her credences change?
Halfer reasoning: On Sunday night, before falling asleep, Beauty knows the coin is fair: P(Heads) = 1/2. When she wakes up, what new information does she have? She was certain she would wake up. Nothing has happened to shift the probability.
Halfers argue that the Thirder position confuses frequency of awakenings with probability of coin outcomes. Yes, 2/3 of awakenings are tails-awakenings, but that doesn't mean the coin is more likely to have landed tails.
Calculate the Probabilities
What if the coin isn't fair? See how the Halfer and Thirder answers differ based on different priors:
P(Heads | You are awake)
33.3%
Thirder reasoning: With prior 50%, accounting for the fact that tails-awakenings are twice as frequent, P(Heads) updates to 33.3%. With a fair coin prior, this gives exactly 1/3.
The Double-Halfer Controversy
Some philosophers go even further than Halfers. What if Beauty is told which day it is?
There is an even more controversial position: the double-halfer.
Double-Halfer
1/2 always"Even if you are told it is Monday, P(Heads) should still be 1/2."
Click to expand and see why double-halfers face difficult objections.
Why This Problem Matters
The Sleeping Beauty problem is not just an academic puzzle. Your answer has implications for some of the biggest questions in philosophy and physics.
Sleeping Beauty is not just a puzzle. It connects to some of the deepest questions in philosophy, physics, and AI safety.
Anthropic Reasoning
How should we reason about our own existence?
The Simulation Argument
Are we living in a simulation?
Many-Worlds Interpretation
How to think about quantum branching?
Decision Theory
How should Beauty bet?
Self-Locating Beliefs
Where and when are you?
Where Do You Stand?
After two decades of debate, there is no consensus. Some of the brightest philosophers remain Halfers; others are convinced Thirders. Both sides have technical arguments in their favor.
The real insight:
Probability theory, as usually formulated, does not tell us how to handle self-locating uncertainty - uncertainty about where and when we are. Sleeping Beauty reveals this gap in our foundations.
“The question is not what credence Beauty should have, but what credence even is.”
Beauty wakes up. She opens her eyes. What should she believe?
Explore Related Paradoxes
The Sleeping Beauty problem connects to deep questions about anthropic reasoning, decision theory, and the foundations of probability.
References: Elga (2000), Lewis (2001), Bostrom (2007)