  We consider the near-critical Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$ and
provide a new probabilistic proof of the fact that, when $p$ is of the form
$p=p(n)=1/n+\lambda/n^{4/3}$ and $A$ is large,
\[\mathbb{P}(|\mathcal{C}_{\max}|>An^{2/3})\asymp
A^{-3/2}e^{-\frac{A^3}{8}+\frac{\lambda A^2}{2}-\frac{\lambda^2A}{2}}\] where
$\mathcal{C}_{\max}$ is the largest connected component of the graph. Our
result allows $A$ and $\lambda$ to depend on $n$. While this result is already
known, our proof relies only on conceptual and adaptable tools such as ballot
theorems, whereas the existing proof relies on a combinatorial formula specific
to Erd\H{o}s-R\'{e}nyi graphs, together with analytic estimates.
