A substantial population of red quasars has been discovered in a complete sample of flat-spectrum radio sources. Dust is the most likely cause of the reddening in this sample. The location of the dust is poorly known, but may either be in the line-of-sight to the quasar, or in the immediate quasar environment. In this paper we are interested in models where the dust is located in the line of sight to the quasar. We calculate the probability distribution of the optical depth in galactic dust as a function of source redshift, using a range of parameters which might describe real galaxies. We compare these results with those found for our sample of radio quasars. If the dust content is unevolving, then it is not possible to account for all the observed reddening in the quasar sample using these models. Our minimum dust model predicts that 15% of background quasars to z ~ 5 will have a line of sight within 5 kpc of a galaxy’s centre, and would therefore be reddened out of B-band flux-limited samples.