  We use a semi-analytical model for the substructure of dark matter haloes to
assess the too-big-to-fail (TBTF) problem. The model accurately reproduces the
average subhalo mass and velocity functions, as well as their halo-to-halo
variance, in N-body simulations. We construct thousands of realizations of
Milky Way (MW) size host haloes, allowing us to investigate the TBTF problem
with unprecedented statistical power. We examine the dependence on host halo
mass and cosmology, and explicitly demonstrate that a reliable assessment of
TBTF requires large samples of hundreds of host haloes. We argue that previous
statistics used to address TBTF suffer from the look-elsewhere effect and/or
disregard certain aspects of the data on the MW satellite population. We devise
a new statistic that is not hampered by these shortcomings, and, using only
data on the 9 known MW satellite galaxies with $V_{\rm max}>15{\rm kms}^{-1}$,
demonstrate that $1.4^{+3.3}_{-1.1}\%$ of MW-size host haloes have a subhalo
population in statistical agreement with that of the MW. However, when using
data on the MW satellite galaxies down to $V_{\rm max}=8{\rm kms}^{-1}$, this
MW consistent fraction plummets to $<5\times10^{-4}$ (at 68% CL). Hence, if it
turns out that the inventory of MW satellite galaxies is complete down to
8km/s, then the maximum circular velocities of MW satellites are utterly
inconsistent with $\Lambda$CDM predictions, unless baryonic effects can
drastically increase the spread in $V_{\rm max}$ values of satellite galaxies
compared to that of their subhaloes.
