  We present the approximated analytic expressions for the muon survival
probability in a $3+1$ mixing scenario in the presence of matter effect using
the S-matrix formalism. We find that all the individual terms contributing to
the muon survival probability can significantly reduce to just three
contributions. The leading order contribution comes from the three flavor muon
survival probability followed by the two sub-leading contributions arising from
active-sterile mixing. Furthermore, to more simplify the results we adopt the
well known series expansion relations about mass-hierarchy parameter $\alpha =
\Delta m^2_{21} / \Delta m^2_{31}$ and the mixing angle $\sin \theta_{13}$ in
the vanishing limit of $\alpha^2$. We discuss the relevance of muon survival
probability to probe the CP and T-violation studies coming from the new
physics. We also compare the analytic relation between vacuum and matter
contributions to the muon survival probability at the leading order. Finally,
we comment on the probability behavior at the various long baselines relevant
to understand the atmospheric-neutrino sector and to resolve the existing
mass-hierarchy problem.
