Bayesian reasoning resolves the two-coin probability paradox
Alice's ambiguous 'left coin' changes probability from 1/2 to 1/3
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The classic two-coin puzzle gets a Bayesian treatment by martinkunev on LessWrong. Alice flips two fair coins and covers them. She then tells Bob 'the left coin is heads'—but 'left' could mean her left or Bob's left. Two reasoning paths emerge: one enumerates possibilities assuming the statement eliminates the irrelevant combination, yielding 1/2. The other says 'one coin is heads' leaves three equally likely outcomes (HH, HT, TH) giving 1/3. Both seem logical but conflict.
The resolution lies in the process generating the statement. Bayesian reasoning requires a likelihood function: what would Alice have said under each possible configuration? If Alice always picks a hand with a head when possible, the probability is 1/3. If she picks a hand randomly, it's 1/2. If she only says 'left is heads' for HH, it's 1. The factual statement 'one coin is heads' is not the same as 'Alice said one coin is heads.' This subtlety matters whenever we update on testimony—the speaker's policy is part of the evidence.
- Two valid-seeming reasoning paths yield 1/2 and 1/3 probabilities for both heads after Alice's ambiguous 'left coin is heads' statement.
- The paradox is resolved by recognizing that Bob's update depends on Alice's unknown policy for choosing what to reveal—a Bayesian likelihood function.
- If Alice always reports a heads when available, the correct probability is 1/3; if she picks a hand at random, it's 1/2—no unique answer without the policy.
Why It Matters
Illustrates that Bayesian updating requires knowing the data-generating process, crucial for AI alignment and decision theory.