  We derive a simple formula characterizing the distribution of the size of the
connected component of a fixed vertex in the Erd\H{o}s-R\'enyi random graph
which allows us to give elementary proofs of some results of Federico, van der
Hofstad, den Hollander and Hulshof as well as Janson and Luczak about the
susceptibility in the subcritical graph and the central limit theorem of Pittel
for the size of the giant component in the supercritical graph.
