Pirate Hat Puzzle: Cooperative Parity Strategy

Ten pirates are each given a hat -- either red or blue -- uniformly at random. Each pirate can see everyone else's hat but not their own. They must simultaneously and independently announce a guess for their own hat color. If every single pirate guesses correctly, they are all freed; if even one is wrong, they are all executed. Before the hats are assigned, the pirates can meet and agree on any strategy they like. No communication is allowed once the hats go on. 1. What is the survival probability of the naive strategy where everyone just guesses randomly? 2. Design an optimal cooperative strategy. What survival probability does it achieve? 3. Prove that no strategy can exceed this probability.