  Given a smooth subscheme of a projective space over a finite field, we
compute the probability that its intersection with a fixed number of
hypersurface sections of large degree is smooth of the expected dimension. This
generalizes the case of a single hypersurface, due to Poonen. We use this
result to give a probabilistic model for the number of rational points of such
a complete intersection. A somewhat surprising corollary is that the number of
rational points on a random smooth intersection of two surfaces in projective
3-space is strictly less than the number of points on the projective line.
