  Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb
F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane
is said to satisfy the Poncelet triangle condition if, considered as conics in
${\mathbf P}^2({\overline{\mathbb F}}_q)$, they intersect transverally and
there exists a triangle inscribed in ${\mathcal A}$ and circumscribed around
${\mathcal B}$. It is shown in this article that a randomly chosen pair of
conics satisfies the triangle condition with asymptotic probability $1/q$. We
also make a conjecture based upon computer experimentation which predicts this
probability for tetragons, pentagons and so on up to enneagons.
