Low-frequency radio surveys are ideal for selecting orientation-independent samples of extragalactic sources because the sample members are selected by virtue of their isotropic steep-spectrum extended emission. We use the new 7C Redshift Survey along with the brighter 3CRR and 6C samples to investigate the fraction of objects with observed broad emission lines - the `quasar fraction' - as a function of redshift and of radio and narrow-emission-line luminosity. We find that the quasar fraction is more strongly dependent upon luminosity (both narrow-line and radio) than it is on redshift. Above a narrow [Oii] emission-line luminosity of log10(L[Oii]/W)>~35 [or radio luminosity log10(L151/WHz-1sr-1)>~ 26.5], the quasar fraction is virtually independent of redshift and luminosity; this is consistent with a simple unified scheme with an obscuring torus with a half-opening angle θtrans~53°. For objects with less luminous narrow lines, the quasar fraction is lower. We show that this is not due to the difficulty of detecting lower luminosity broad emission lines in a less luminous, but otherwise similar, quasar population. We discuss evidence which supports at least two probable physical causes for the drop in quasar fraction at low luminosity: (i) a gradual decrease in θtrans and/or a gradual increase in the fraction of lightly reddened (0<~AV<~5) lines of sight with decreasing quasar luminosity; and (ii) the emergence of a distinct second population of low-luminosity radio sources which, like M87, lack a well-fed quasar nucleus and may well lack a thick obscuring torus.