  We investigate the extent to which the probabilistic properties of a chaotic
scattering system with dissipation can be understood from the properties of the
dissipation-free system. For large energies $E$, a fully chaotic scattering
leads to an exponential decay of the survival probability $P(t) \sim e^{-\kappa
t}$ with an escape rate $\kappa$ that decreases with $E$. Dissipation
$\gamma>0$ leads to the appearance of different finite-time regimes in $P(t)$.
We show how these different regimes can be understood for small $\gamma\ll 1$
and $t\gg 1/\kappa_0$ from the effective escape rate
$\kappa_\gamma(t)=\kappa_0(E(t))$ (including the non-hyperbolic regime) until
the energy reaches a critical value $E_c$ at which no escape is possible. More
generally, we argue that for small dissipation $\gamma$ and long times $t$ the
surviving trajectories in the dissipative system are distributed according to
the conditionally invariant measure of the conservative system at the
corresponding energy $E(t)<E(0)$. Quantitative predictions of our general
theory are compared with numerical simulations in the Henon-Heiles model.
