We present a Markov-Chain Monte-Carlo orbit computation method evolving from statistical ranging, for poorly observed single-apparition asteroids with two or more observations. We examine the Bayesian a posteriori probability density of the orbital elements using methods that map a volume of orbits in the orbital-element phase space. In particular, we use the MCMC method to sample the phase space in an unbiased way. We study the speed of convergence and also the efficiency of the new method for the initial orbit computation problem. We present the results of the MCMC ranging method applied to three objects from di-erent dynamical groups. We conclude that the method is applicable to initial orbit computation for near-Earth, main-belt, and transneptunian objects. Similarly to ranging, the MCMC variant could also be applied to a variety of different topics such as rigorous ephemeris prediction, orbital-element-distribution studies for transneptunian objects, the computation of invariant collision probabilities between near-Earth objects and the Earth, detecting linkages between astrometric asteroid observations within an apparition as well as between apparitions, and in the rigorous analysis of the impact of orbital arc length and/or astrometric uncertainty on the uncertainty of the resulting orbits.