  Fix a positive integer $n$ and a finite field $\mathbb F_q$. We study the
joint distribution of the rank of $E$, the $n$-Selmer group of $E$, and the
$n$-torsion in the Tate-Shafarevich group of $E$ as $E$ varies over elliptic
curves of fixed height $d \geq 2$ over $\mathbb F_q(t)$. We compute this joint
distribution in the large $q$ limit. We also show that the "large $q$, then
large height" limit of this distribution agrees with the one predicted by
Bhargava-Kane-Lenstra-Poonen-Rains.
