Fibered commensurability and arithmeticity of random mapping tori

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 the 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 the random walk gives a mapping class with an arithmetic mapping torus decays exponentially.