  In the polluted bootstrap percolation model, vertices of the cubic lattice
$\mathbb{Z}^3$ are independently declared initially occupied with probability
$p$ or closed with probability $q$. Under the standard (respectively, modified)
bootstrap rule, a vertex becomes occupied at a subsequent step if it is not
closed and it has at least $3$ occupied neighbors (respectively, an occupied
neighbor in each coordinate). We study the final density of occupied vertices
as $p,q\to 0$. We show that this density converges to $1$ if $q \ll p^3(\log
p^{-1})^{-3}$ for both standard and modified rules. Our principal result is a
complementary bound with a matching power for the modified model: there exists
$C$ such that the final density converges to $0$ if $q > Cp^3$. For the
standard model, we establish convergence to $0$ under the stronger condition
$q>Cp^2$.
