We study the properties of the population of optically dark events present in a carefully selected complete sample of bright Swift long gamma-ray bursts. The high level of completeness in redshift of our sample (52 objects out of 58) allows us to establish the existence of a genuine dark population, and we are able to estimate the maximum fraction of dark burst events (∼30 per cent) expected for the whole class of long gamma-ray burst. The redshift distribution of this population of dark bursts is similar to that of the whole sample. Interestingly, the rest-frame X-ray luminosity (and the de-absorbed X-ray flux) of the subclass of dark bursts is slightly higher than the average luminosity of the non-dark events. At the same time, the prompt properties do not differ and the optical flux of dark events is at the lower tail of the optical flux distribution, corrected for Galactic absorption. All these properties suggest that dark bursts events generate in much denser environments with respect to normal bright events. We can therefore exclude the high-z and the low-density scenarios and conclude that the major cause of the origin of optically dark events is the dust extinction.