We carefully reconsider the problem of classifying broad-absorption line quasars (BALQSOs) and derive a new, unbiased estimate of the intrinsic BALQSO fraction from the Sloan Digital Sky Survey (SDSS) DR3 quasi-stellar object (QSO) catalogue. We first show that the distribution of objects selected by the so-called ‘absorption index’ (AI) is clearly bimodal in log AI, with only one mode corresponding to definite BALQSOs. The surprisingly high BALQSO fractions that have recently been inferred from AI-based samples are therefore likely to be overestimated. We then present two new approaches to the classification problem that are designed to be more robust than the AI, but also more complete than the traditional ‘balnicity index’ (BI). Both approaches yield observed BALQSO fractions around 13.5 per cent, while a conservative third approach suggests an upper limit of 18.3 per cent. Finally, we discuss the selection biases that affect our observed BALQSO fraction. After correcting for these biases, we arrive at our final estimate of the intrinsic BALQSO fraction. This is fBALQSO= 0.17 ± 0.01 (stat) ± 0.03 (sys) with an upper limit of fBALQSO≃ 0.23. We conclude by pointing out that the bimodality of the log AI distribution may be evidence that the BAL-forming region has clearly delineated physical boundaries.