A reason I think may be is that the entropy $H(X)$ for the random variables $X_1,X_2, ..,X_n$ has a bound $H(X) \leq logn$, where equality holds for $p_i = 1/n, ~ \forall i=1,2,..,n$ e.g when tossing a fair coin. If the trial of error is $p=1/2$ we get maximum entropy, so there's no reason to work for $p> 1/2$