  We prove that the expected area of the amoeba of a complex plane curve of
degree $d$ is less than $\displaystyle{3\ln(d)^2/2+9\ln(d)+9}$ and once
rescaled by $\ln(d)^2$, is asymptotically bounded from below by $3/4$. In order
to get this lower bound, given disjoint isometric embeddings of a bidisc of
size $1/\sqrt{d}$ in the complex projective plane, we lower estimate the
probability that one of them is a submanifold chart of a complex plane curve.
It exponentially converges to one as the number of bidiscs grow to $+\infty$.
