What is the best approximation for majority vote?
If $p_v &gt; p_i$ for all $i \ne v$, then

$$
\mathrm{Pr}[\mbox{outcome is different from $v$}] \leq \min_T \left(Pr[B(N, p_v) \leq T] + Pr[\forall i \ne v \quad B(N, p_i) \geq T]\right),
$$

where $B(n, p)$ is binomial distribution, and $T$ is an arbitrary threshold. Plugging $T = N (p_v + \max_{i \ne v} p_v) / 2$ and using Chernoff bounds, one can upper-bound this probability as $e^{-\Omega(N)}$.

Of course, if $p_v$ is not maximum, then you get the opposite picture. With overwhelming probability $v$ is not outcome.