  Let a lattice gas of constant density, described by the symmetric simple
exclusion process, be brought in contact with a "target": a spherical absorber
of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate
the probability ${\mathcal P}(T)$ that no gas particle hits the target until a
long but finite time $T$. We also find the most likely gas density history
conditional on the non-hitting. The results depend on the dimension of space
$d$ and on the rescaled parameter $\ell=R/\sqrt{D_0T}$, where $D_0$ is the gas
diffusivity. For small $\ell$ and $d>2$, ${\mathcal P}(T)$ is determined by an
exact stationary solution of the MFT equations that we find. For large $\ell$,
and for any $\ell$ in one dimension, the relevant MFT solutions are
non-stationary. In this case $\ln {\mathcal P}(T)$ scales differently with
relevant parameters, and it also depends on whether the initial condition is
random or deterministic. The latter effects also occur if the lattice gas is
composed of non-interacting random walkers. Finally, we extend the formalism to
a whole class of diffusive gases of interacting particles.
