  Let X be a smooth complex projective manifold of dimension n equipped with an
ample line bundle L and a rank k holomorphic vector bundle E. We assume that 0<
k <=n, that X, E and L are defined over the reals and denote by RX the real
locus of X. Then, we estimate from above and below the expected Betti numbers
of the vanishing loci in RX of holomorphic real sections of E tensored with
L^d, where d is a large enough integer. Moreover, given any closed connected
codimension k submanifold S of R^n with trivial normal bundle, we prove that a
real section of E tensored with L^d has a positive probability, independent of
d, to contain around the square root of d^n connected components diffeomorphic
to S in its vanishing locus.
