  We consider random lattices taken from the general symplectic ensemble and
count the number of lattice points of a typical lattice in nested families
$B_t$ of certain Borel sets. Our main result is that for almost every general
symplectic lattice, the discrepancy $D(\Lambda,B_t)$ of the lattice point count
with respect to the volumes is $O(vol(B_t)^{-\delta})$. This extends work of W.
Schmidt who gave similar discrepancy bounds for the space of all lattices in
$\mathbb{R}^n$.
