We study the probability for a random line to intersect a givenplane curve, defined over a finite field, in a given number of pointsdefined over the same field. In particular, we focus on the limits ofthese probabilities under successive finite field extensions. Veronesemaps allow us to compute similar probabilities of intersection between agiven curve and random curves of given degree. Finally, we use Bertini'sTheorem to compute the probability that a random linear subspace of theright dimension intersects X in a given number of points.