  In this paper, we construct a natural probability measure on the space of
real branched coverings from a real projective algebraic curve $(X,c_X)$ to the
projective line $(\mathbb{C}\mathbb{P}^1,\textrm{conj})$. We prove that the
space of degree $d$ real branched coverings having "many" real branched points
(for example more than $\sqrt{d}^{1+\alpha}$, for any $\alpha>0$) has
exponentially small measure. In particular, maximal real branched coverings,
that is real branched coverings such that all the branched points are real, are
exponentially rare.
