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 log L_[OII] > 35 W (or radio luminosity log L_151 > 26.5 W/Hz/sr), 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 theta_trans approx 53 degrees. 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 theta_trans and/or a gradual increase in the fraction of lightly-reddened (0 < A(V) < 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.