In this paper we apply our Monte-Carlo photometric-redshift technique, introduced in paper I (Hughes et al. 2002), to the multi-wavelength data available for 77 galaxies selected at 850um and 1.25mm. We calculate a probability distribution for the redshift of each galaxy, which includes a detailed treatment of the observational errors and uncertainties in the evolutionary model. The cumulative redshift distribution of the submillimetre galaxy population that we present in this paper, based on 50 galaxies found in wide-area SCUBA surveys, is asymmetric, and broader than those published elsewhere, with a significant high-z tail for some of the evolutionary models considered. Approximately 40 to 90 per cent of the sub-mm population is expected to have redshifts in the interval 2 < z < 4. Whilst this result is completely consistent with earlier estimates for the sub-mm galaxy population, we also show that the colours of many (< 50 per cent) individual sub-mm sources, detected only at 850um with non-detections at other wavelengths, are consistent with those of starburst galaxies that lie at extreme redshifts, z > 4. Spectroscopic confirmation of the redshifts, through the detection of rest-frame FIR--mm wavelength molecular transition-lines, will ultimately calibrate the accuracy of this technique. We use the redshift probability distribution of HDF850.1 to illustrate the ability of the method to guide the choice of possible frequency tunings on the broad-band spectroscopic receivers that equip the large aperture single-dish mm and cm-wavelength telescopes.
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