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 [O ii] emission-line luminosity of log10(L[O ii]/W)≳35 [or radio luminosity log10(L151/W Hz−1 sr−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.