  We find that the fraction of classical Broad Absorption Line quasars
(BALQSOs) among the FIRST radio sources in the Sloan Data Release 3, is
20.5^{+7.3}_{-5.9}% at the faintest radio powers detected (L_{\rm 1.4
GHz}~10^{32} erg/s), and rapidly drops to <8% at L_{\rm 1.4 GHz}~3*10^{33}
erg/s. Similarly, adopting the broader Absorption Index (AI) definition of
Trump et al. (2006) we find the fraction of radio BALQSOs to be
44^{+8.1}_{-7.8}% reducing to 23.1^{+7.3}_{-6.1}% at high luminosities. While
the high fraction at low radio power is consistent with the recent near-IR
estimates by Dai et al. (2008), the lower fraction at high radio powers is
intriguing and confirms previous claims based on smaller samples. The trend is
independent of the redshift range, the optical and radio flux selection limits,
or the exact definition of a radio match. We also find that at fixed optical
magnitude, the highest bins of radio luminosity are preferentially populated by
non-BALQSOs, consistent with the overall trend. We do find, however, that those
quasars identified as AI-BALQSOs but \emph{not} under the classical definition,
do not show a significant drop in their fraction as a function of radio power,
further supporting independent claims for which these sources, characterized by
lower equivalent width, may represent an independent class with respect to the
classical BALQSOs. We find the balnicity index, a measure of the absorption
trough in BALQSOs, and the mean maximum wind velocity to be roughly constant at
all radio powers. We discuss several plausible physical models which may
explain the observed fast drop in the fraction of the classical BALQSOs with
increasing radio power, \emph{although no one is entirely satisfactory}.
(abridged).
