Poncelet Triangles and Tetragons over Finite Fields

In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathcal{A},\mathcal{B})$, with $\mathcal{A}$ smooth or singular and $\mathcal{B}$ smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in $\mathcal{A}$ and circumscribed about $\mathcal{B}$. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally.