Proportion of blocking curves in a pencil

Let $\mathcal{L}$ be a pencil of plane curves defined over ${\mathbb{F}}_q$ with no ${\mathbb{F}}_q$-points in its base locus. We investigate the number of curves in $\mathcal{L}$ whose ${\mathbb{F}}_q$-points form a blocking set. When the degree of the pencil is allowed to grow with respect to $q$, we show that the geometric problem can be translated into a purely combinatorial problem about disjoint blocking sets. We also study the same problem when the degree of the pencil is fixed.