  The statistics of Einstein radii for a sample of strong lenses can provide
valuable constraints on the underlying mass distribution. The correct
interpretation, however, relies critically on the modelling of the selection of
the sample, which has proven to be a limiting factor. This may change thanks to
upcoming uniform high-resolution imaging surveys that cover a large fraction of
the sky, because they can provide complete lens samples, with well understood
selection criteria. To explore how the observed distribution of Einstein radii
depends on the galaxy properties, we simulated a realistic complete sample of
strong lenses, predicting a number density of lenses of about 2.5 deg$^{-2}$
for a \Euclid-like setup. Such data can break the degeneracy between the
stellar initial mass function (IMF) and the inner slope of the density profile
of dark matter, without having to rely on additional information from stellar
dynamics. We found that a survey covering only 50 deg$^2$ can already provide
tight constraints: assuming that the cosmology is known, the dark matter slope
is recovered with an uncertainty of $3.5\%$, while the uncertainty in the ratio
between the true stellar mass and that inferred from stellar population
modelling was found to be $10\%$. These findings highlight the potential of
this method when applied to samples of lenses with well-understood selection
functions.
