  We present an analysis of the detection fraction and the number counts of
radio sources imaged with Very Long Baseline Interferometry (VLBI) at 1.4 GHz
as part of the mJIVE-20 survey. From a sample of 24,903 radio sources
identified by FIRST, 4,965 are detected on VLBI-scales, giving an overall
detection fraction of $19.9\pm2.9~$per cent. However, we find that the
detection fraction falls from around 50 per cent at a peak surface brightness
of $80~mJy~beam^{-1}$ in FIRST to around 8 per cent at the detection limit,
which is likely dominated by the surface brightness sensitivity of the VLBI
observations, with some contribution from a change in the radio source
population. We also find that compactness at arcsec-scales is the dominant
factor in determining whether a radio source is detected with VLBI, and that
the median size of the VLBI-detected radio sources is 7.7 mas. After correcting
for the survey completeness and effective sky area, we determine the slope of
the differential number counts of VLBI-detected radio sources with flux
densities $S_{\rm 1.4~GHz} > 1~mJy$ to be $\eta_{\rm VLBI} = -1.74\pm 0.02$,
which is shallower than in the cases of the FIRST parent population ($\eta_{\rm
FIRST} = -1.77\pm 0.02$) and for compact radio sources selected at higher
frequencies ($\eta_{\rm JBF} = -2.06\pm 0.02$). From this, we find that all-sky
($3\pi~sr$) surveys with the EVN and the VLBA have the potential to detect
$(7.2\pm0.9)\times10^{5}$ radio sources at mas-resolution, and that the density
of compact radio sources is sufficient (5.3~deg$^{-2}$) for in-beam phase
referencing with multiple sources (3.9 per primary beam) in the case of a
hypothetical SKA-VLBI array.
