  Extensive Monte-Carlo simulations were performed to evaluate the excess
number of clusters and the crossing probability function for three-dimensional
percolation on the simple cubic (s.c.), face-centered cubic (f.c.c.), and
body-centered cubic (b.c.c.) lattices. Systems L x L x L' with L' >> L were
studied for both bond (s.c., f.c.c., b.c.c.) and site (f.c.c.) percolation. The
excess number of clusters $\tilde {b}$ per unit length was confirmed to be a
universal quantity with a value $\tilde {b} \approx 0.412$. Likewise, the
critical crossing probability in the L' direction, with periodic boundary
conditions in the L x L plane, was found to follow a universal exponential
decay as a function of r = L'/L for large r. Simulations were also carried out
to find new precise values of the critical thresholds for site percolation on
the f.c.c. and b.c.c. lattices, yielding $p_c(f.c.c.)= 0.199 236 5 \pm 0.000
001 0$, $p_c(b.c.c.)= 0.245 961 5\pm 0.000 001 0$.
