  We study the probability for a random line to intersect a given plane curve,
defined over a finite field, in a given number of points defined over the same
field. In particular, we focus on the limits of these probabilities under
successive finite field extensions. Supposing absolute irreducibility for the
curve, we show how a variant of Chebotarev density theorem for function fields
can be used to prove the existence of these limits, and to compute them under a
mildly stronger condition, known as simple tangency. Partial results have
already appeared in the literature, and we propose this work as an introduction
to the use of Chebotarev theorem in the context of incidence geometry. Finally,
Veronese maps allow us to compute similar probabilities of intersection between
a given curve and random curves of given degree.
