  We consider a random walk on the mapping class group of a surface of finite
type. We assume that the random walk is determined by a probability measure
whose support is finite and generates a non-elementary subgroup $H$. We further
assume that $H$ is not consisting only of lifts with respect to any one
covering. Then we prove that the probability that such a random walk gives a
non-minimal mapping class in its fibered commensurability class decays
exponentially. As an application of the minimality, we prove that for the case
where a surface has at least one puncture, the probability that a random walk
gives mapping classes with arithmetic mapping tori decays exponentially. We
also prove that a random walk gives rise to asymmetric mapping tori with
exponentially high probability for closed case.
