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 (L1.4 GHz ~ 1032 erg s−1), and rapidly drops to 8% at L1.4 GHz ~ 3 × 1033 erg s−1. Similarly, adopting the broader absorption index (AI) definition of Trump et al., 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., 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 not under the classical definition do not show a significant drop in their fraction as a function of radio power, further supporting independent claims that these sources, characterized by lower equivalent width, may represent an independent class from 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, although none is entirely satisfactory. A strictly evolutionary model for the BALQSO and radio emission phases requires a strong fine-tuning to work, while a simple geometric model, although still not capable of explaining polar BALQSOs and the paucity of FRII BALQSOs, is statistically successful in matching the data if part of the apparent radio luminosity function is due to beamed, non-BALQSOs.