3 people have each been assigned \(H\) or \(T\) on their foreheads, based on the results of tossing a fair coin. Each member can see each others' letters but not their own. Their common goal is to win a game with the following rules:

  1. They must all make a statement at the same time, and no communication is allowed beforehand.
  2. Each member's statement can either be a guess of their own letter ("H" or "T") or "pass".
  3. They win if at least one person guesses correctly and no one guesses incorrectly.

With everyone applying the optimal strategy, what is their probability of winning?


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