Two Players, Five Colours
Alice and Bob agree on a strategy, after which each receives a hat in one of five colours, chosen uniformly and independently. Each sees the other's hat but not their own, and no communication is allowed once the hats are on. Simultaneously, each must name a colour as their guess for their own hat (no passing). The pair wins if at least one guess is correct. What is the largest possible probability of winning? Give a reduced fraction.
Related problems
Source: Small case of the classic hat-guessing game with many colours, in wide circulation; no traceable original source. Statement written for AxiomIQ.