  Given a graph $G$ and some initial labelling $\sigma : V(G) \to \{Red,
Blue\}$ of its vertices, the \textit{majority dynamics model} is the
deterministic process where at each stage, every vertex simultaneously replaces
its label with the majority label among its neighbors (remaining unchanged in
the case of a tie). We prove---for a wide range of parameters---that if an
initial assignment is fixed and we independently sample an Erd\H{o}s--R\'enyi
random graph, $G_{n,p}$, then after one step of majority dynamics, the number
of vertices of each label follows a central limit law. As a corollary, we
provide a strengthening of a theorem of Benjamini, Chan, O'Donnell, Tamuz, and
Tan about the number of steps required for the process to reach unanimity when
the initial assignment is also chosen randomly. Moreover, suppose there are
initially three more red vertices than blue. In this setting, we prove that if
we independently sample the graph $G_{n,1/2}$, then with probability at least
$51\%$, the majority dynamics process will converge to every vertex being red.
This improves a result of Tran and Vu who addressed the case that the initial
lead is at least 10.
