  The hole probability, i.e., the probability that a region is void of
particles, is a benchmark of correlations in many body systems. We compute
analytically this probability $P(R)$ for a spherical region of radius $R$ in
the case of $N$ noninteracting fermions in their ground state in a
$d$-dimensional trapping potential. Using a connection to the Laguerre-Wishart
ensembles of random matrices, we show that, for large $N$ and in the bulk of
the Fermi gas, $P(R)$ is described by a universal scaling function of $k_F R$,
for which we obtain an exact formula ($k_F$ being the local Fermi wave-vector).
It exhibits a super exponential tail $P(R)\propto e^{- \kappa_d (k_F R)^{d+1}}$
where $\kappa_d$ is a universal amplitude, in good agreement with existing
numerical simulations. When $R$ is of the order of the radius of the Fermi gas,
the hole probability is described by a large deviation form which is not
universal and which we compute exactly for the harmonic potential. Similar
results also hold in momentum space.
