  In [5], Manjul Bhargava and Benedict Gross considered the family of
hyperelliptic curves over $\Q$ having a fixed genus and a marked rational
Weierstrass point. They showed that the average size of the 2-Selmer group of
the Jacobians of these curves, when ordered by height, is 3. In this paper, we
consider the family of hyperelliptic curves over $\Q$ having a fixed genus and
a marked rational non-Weierstrass point. We show that when these curves are
ordered by height, the average size of the 2-Selmer group of their Jacobians is
6. This yields an upper bound of 5/2 on the average rank of the Mordell-Weil
group of the Jacobians of these hyperelliptic curves.
  Finally using an equidistribution result, we modify the techniques of [16] to
conclude that as $g$ tends to infinity, a proportion tending to 1 of these
monic even-degree hyperelliptic curves having genus $g$ have exactly two
rational points - the marked point at infinity and its hyperelliptic conjugate.
