Using the Bright Star Catalog (BSC5P) and NVSS Source Catalog as
an example we find that while the BSC5P entries are
spread over the entire 4 pi steradians of the sky
(= 41,252.961 square degrees = 1.485107 x 10^8 square arcminutes),
the NVSS only covers the 82.1% of the sky North of -40 declination which is
easily visible to the VLA. A separate Browse query of the BSC5P table reveals
that only 7240 of the 9110 BSC5P entries are in this part of the sky. Thus,
when you cross-correlate the BSC5P table against the NVSS table with a 0.25'
cross-correlation radius, you are actually searching an
area of the sky = 7240 x pi x (0.25)^2 = 1,421.57 square arcminutes.
The surface density of NVSS entries is 1,773,484/(0.821 x 1.485107 x 10^8)
sources per square arcminute, and thus multiplying the area searched by
the latter catalog's surface density, you get 20.7 random matches expected.
Assuming Poissonian uncertainties, the uncertainty in this estimate is
20.7^0.5 = 4.5, i.e., there are likely 16.2 - 25.2 random matches.
If you actually perform this particular cross-correlation, you will find
that 43 of the BSC5P entries have matches within the cross-correlation radius
with 41 NVSS entries, which is a significantly larger number than the expected
number of random matches.