  Let $\Gamma_{g,1}^m$ be the mapping class group of the orientable surface
$\Sigma_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$
permutable punctures; when $m=0$ we omit it from the notation. Let
$\beta_{m}(\Sigma_{g,1})$ be the braid group on $m$ strands of the surface
$\Sigma_{g,1}$.
  We prove that $H_*(\Gamma_{g,1}^m;\mathbb{Z}_2)\cong
H_*(\Gamma_{g,1};H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2))$. The main
ingredient is the computation of $H_*(\beta_{m}(\Sigma_{g,1});\mathbb{Z}_2)$ as
a symplectic representation of $\Gamma_{g,1}$.
