Given a graph $G$ of $n$ vertices, where each vertex is initially attached an opinion of either red or blue. We investigate a random process known as the Best-of-three voting. In this process, at each time step, every vertex chooses three neighbours at random and adopts the majority colour. We study this process for a class of graphs with minimum degree $d=n\alpha$,, where $\alpha =\Omega ((\log \log n)-1)$. We prove that if initially each vertex is red with probability greater than $1/2+\delta$, and blue otherwise, where $\delta \ge (\log d)-C$ for some $C>0$, then with high probability this dynamic reaches a final state where all vertices are red within $O(\log \log n)+O(\log (\delta -1))$ steps.