  We compute $E_G (\prod_i \tr(g^{\lambda_i}))$, where $G=Sp(2n)$ or $SO(m)
(m=2n, 2n+1)$ with Haar measure. This was first obtained by Persi Diaconis and
Mehrdad Shahshahani, but our proof is more self-contained and gives a
combinatorial description for the answer. We also consider how averages of
general symmetric functions $E_G f_n$ are affected when we introduce a Weyl
character $\chi^G_\lambda$ into the integrand.
  We show that the value of $E_G \chi^G_\lambda f_n / E_G f_n$ approaches a
constant for large $n$. More surprisingly, the ratio we obtain only changes
with $f_n$ and $\lambda$ and is independent of the Cartan type of $G$. Even in
the unitary case, Daniel Bump and Persi Diaconis have obtained the same ratio.
Finally, those ratios can be combined with asymptotics for $E_G f_n$ due to
Kurt Johansson and provide asymptotics for $E_G \chi^G_\lambda f_n$.
