KöMaL 2024 A874

Nyihaha and Bruhaha are two neighbouring islands, both having $n$ inhabitants. On island Nyihaha every inhabitant is either a Knight or a Knave. Knights always tell the truth and Knaves always lie. The inhabitants of island Bruhaha are normal people, who can choose to say whatever they want. When a visitor arrives on any of the two islands, the following ritual is performed: every inhabitant points randomly to another inhabitant (independently from each other with uniform distribution), and says He is a Knight'' or He is a Knave''. On island Nyihaha , Knights have to tell the truth and Knaves have to lie. On island Bruhaha every inhabitant says either He is a Knight'' or He is a Knave'' with probability $1/2$ independently from each other. Sinbad arrives on island Bruhaha , but he does not know whether he is on island Nyihaha or island Bruhaha . Let $p_n$ denote the probability that after observing the ritual he can rule out being on island Nyihaha . Is it true that $p_n \to 1$ if $n \to \infty$?