An expression for the probability of transmission of an electron from an ordinary metal into a medium with a finite quasiparticle lifetime is derived. The derivation is based on using a certain matrix, analogous to the transmission matrix for the elastic transport case, assigned to each region of the sample being investigated. This matrix is determined by the functional shapes of the electronic reciprocal-effective-mass tensor, the one-particle potential energy and the retarded self-energy in a given sample region, as well as by the energy and direction of motion of the incident electron. As an example of an application, the differential conductance is calculated for a tunnelling contact between an ordinary metal and an anisotropic marginal-Fermi-liquid metal for various characters of the contact interlayer zone.