  We investigate strong gravitational lensing in the concordance $\Lambda$CDM
cosmology by carrying out ray-tracing along past light cones through the
Millennium Simulation, the largest simulation of cosmic structure formation
ever carried out. We extend previous ray-tracing methods in order to take full
advantage of the large volume and the excellent spatial and mass resolution of
the simulation. As a function of source redshift we evaluate the probability
that an image will be highly magnified, will be highly elongated or will be one
of a set of multiple images. We show that such strong lensing events can almost
always be traced to a single dominant lensing object and we study the mass and
redshift distribution of these primary lenses. We fit analytic models to the
simulated dark halos in order to study how our optical depth measurements are
affected by the limited resolution of the simulation and of the lensing planes
that we construct from it. We conclude that such effects lead us to
underestimate total strong-lensing cross sections by about 15 percent. This is
smaller than the effects expected from our neglect of the baryonic components
of galaxies. Finally we investigate whether strong lensing is enhanced by
material in front of or behind the primary lens. Although strong lensing
lines-of-sight are indeed biased towards higher than average mean densities,
this additional matter typically contributes only a few percent of the total
surface density.
