  In this paper, we develop a way to extract information about a random walk
associated with a typical Thurston's construction. We first observe that a
typical Thurston's construction entails a free group of rank 2. We also present
a proof of the spectral theorem for random walks associated with Thurston's
construction that have finite second moment with respect to the Teichm\"uller
metric. Its general case was remarked by Dahmani and Horbez. Finally, under a
hypothesis not involving moment conditions, we prove that random walks
eventually become pseudo-Anosov.
  As an application, we first discuss a random analogy of Kojima and McShane's
estimation of the hyperbolic volume of a mapping torus with pseudo-Anosov
monodromy. As another application, we discuss non-probabilistic estimations of
stretch factors from Thurston's construction and the powers for Salem numbers
to become the stretch factors of pseudo-Anosovs from Thurston's construction.
