  We investigate first-passage percolation on the lattice $\Z^d$ for dimensions
$d \geq 2$. Each edge $e$ of the graph is assigned an independent copy of a
non-negative random variable $\tau$. We only assume $\P[\tau=0]0$ is explicit)
for the probability of having two disjoint geodesics between two pairs of
neighbouring vertices at distance $n$. Additionally, under more specific
assumptions on the distribution of $\tau$, we obtain similar lower bounds for
the probability of having two disjoint geodesics (except for their starting and
ending points) between the same two vertices.
