Each individual still has a ~50% chance of being unable to find their name using the query sequence protocol. By agreeing to the same query sequence their success modes and failure modes are now linked. If there's a cycle of 51 or greater, all of those individuals in that cycle are guaranteed to fail, while if they're in a cycle of 50 or less, all of those individuals are guaranteed to succeed.

What this protocol does is make it so each individual failure mode is much more likely to overlap with each other failure mode. Correlating the performance of individuals when you have a one fail all fail scenario is a common solution to this type of problem.