  Bootstrap percolation on a graph with infection threshold $r\in \mathbb{N}$
is an infection process, which starts from a set of initially infected vertices
and in each step every vertex with at least $r$ infected neighbours becomes
infected. We consider bootstrap percolation on the binomial random graph
$G(n,p)$, which was investigated among others by Janson, \L uczak, Turova and
Valier (2012). We improve their results by strengthening the probability bounds
for the number of infected vertices at the end of the process.
