  We consider systems of two specific piecewise linear homeomorphisms of the
unit interval, so called the Alsed\`a-Misiurewicz systems, and investigate the
basic properties of Markov chains which arise when these two transformations
are applied randomly with probabilities depending on the point of the interval.
Though this iterated function system is not contracting in average and known
methods do not apply, stability and the strong law of large numbers are proven.
