 The first conjunct (‘is greater than each individual’s probability’) is the non-asymptotic conclusion, the second (‘converges to 1’) the asymptotic conclusion. One can further show that, if the two states of the world have an equal prior probability (i.e., \(Pr(X = 1) = Pr(X = -1) = 1/2)\), majority rule is the most reliable of all aggregation rules, maximizing \(Pr(V = X)\) (e.g., Ben-Yashar and Nitzan 1997).