A useful result about leftmost and rightmost paths in two dimensional bond
percolation is proved. This result was introduced without proof in \cite{G} in
the context of the contact process in continuous time. As discussed here, it
also holds for several related models, including the discrete time contact
process and two dimensional site percolation. Among the consequences are a
natural monotonicity in the probability of percolation between different sites
and a somewhat counter-intuitive correlation inequality