A fascinating thought experiment challenges the old saying that two heads are better than one. Imagine a coin flip where a friend named Alice lies 20% of the time. By trusting her report, you can guess the coin's outcome correctly 80% of the time. This establishes a baseline for how unreliable information works in a simple scenario.

The puzzle deepens when Bob joins the game. He also lies 20% of the time and decides independently. Intuitively, you might think his testimony would improve your odds, but a simulation proves otherwise. Your accuracy remains stuck at exactly 80%, revealing a counterintuitive statistical truth about how independent errors interact.

The reason for this plateau is the new possibility of a tie. When Alice and Bob disagree, their conflicting reports cancel each other out, leaving you no better off than random chance. This lost ground perfectly offsets any extra confidence you gain when they happen to agree. However, adding a third friend, Charlie, breaks the tie and boosts accuracy to 90%.

This phenomenon has real-world implications for decision-making systems. It explains why a simple majority vote fails with an even number of participants, a concept related to Condorcet's jury theorem. Understanding when adding more advisors helps—and when it creates unresolvable conflict—is critical for designing effective committees, judicial panels, and corporate boards that must reach a consensus.