Seven Hats, One Strategy

Seven players agree on a strategy, after which each receives a hat that is red or blue, chosen independently by a fair coin. Each player sees the six other hats but not their own, and no communication is allowed once the hats are on. Simultaneously and silently, each player either guesses their own colour or passes. The team wins if at least one player guesses and every guessing player is correct. What is the largest probability of winning over all strategies? Give a reduced fraction.

• parity

Source: Well-known hat-guessing puzzle whose optimal strategy comes from the Hamming code; folklore in the recreational-mathematics literature. Statement written for AxiomIQ.