  Suppose that an infinite lattice gas of constant density $n_0$, whose
dynamics are described by the symmetric simple exclusion process, is brought in
contact with a spherical absorber of radius $R$. Employing the macroscopic
fluctuation theory and assuming the additivity principle, we evaluate the
probability distribution ${\mathcal P}(N)$ that $N$ particles are absorbed
during a long time $T$. The limit of $N=0$ corresponds to the survival problem,
whereas $N\gg \bar{N}$ describes the opposite extreme. Here $\bar{N}=4\pi R D_0
n_0 T$ is the \emph{average} number of absorbed particles (in three
dimensions), and $D_0$ is the gas diffusivity. For $n_0\ll 1$ the exclusion
effects are negligible, and ${\mathcal P}(N)$ can be approximated, for not too
large $N$, by the Poisson distribution with mean $\bar{N}$. For finite $n_0$,
${\mathcal P}(N)$ is non-Poissonian. We show that $-\ln{\mathcal P}(N) \simeq
n_0 N^2/\bar{N}$ at $N\gg \bar{N}$. At sufficiently large $N$ and $n_0<1/2$ the
most likely density profile of the gas, conditional on the absorption of $N$
particles, is non-monotonic in space. We also establish a close connection
between this problem and that of statistics of current in finite open systems.
