  We give a short determination of the distribution of the number of
$\F_q$-rational points on a random trigonal curve over $\F_q$, in the limit as
the genus of the curve goes to infinity. In particular, the expected number of
points is $q+2-\frac{1}{q^2+q+1}$, contrasting with recent analogous results
for cyclic $p$-fold covers of $\mathbb P^1$ and plane curves which have an
expected number of points of $q+1$ (by work of Kurlberg, Rudnick, Bucur, David,
Feigon and Lal\'in) and curves which are complete intersections which have an
expected number of points $<q+1$ (by work of Bucur and Kedlaya). We also give a
conjecture for the expected number of points on a random $n$-gonal curve with
full $S_n$ monodromy based on function field analogs of Bhargava's heuristics
for counting number fields.
