  We consider transport through a non-Hermitian conductor connected to a pair
of Hermitian leads and analyze the underlying non-Hermitian scattering problem.
In a typical non-Hermitian system, such as a Hatano--Nelson-type asymmetric
hopping model, the continuity of probability and probability current is broken
at a local level. As a result, the notion of transmission and reflection
probabilities becomes ill-defined. Instead of these probabilities, we introduce
the injection rate $R_{\rm I}=1-|{\cal R}|^2$ and the transmission rate $R_{\rm
T}=|{\cal T}|^2$ as relevant physical quantities, where ${\cal T}$ and ${\cal
R}$ are the transmission and reflection amplitudes, respectively. In a generic
non-Hermitian case, $R_{\rm I}$ and $R_{\rm T}$ have independent information.
We provide a modified continuity equation in terms of incoming and outgoing
currents, from which we derive a global probability conservation law that
relates $R_{\rm I}$ and $R_{\rm T}$. We have tested the usefulness of our
probability conservation law in the interpretation of numerical results for
non-Hermitian localization and delocalization phenomena.
