Combinatorial billiards is a new topic that concerns rigid and discretized billiard systems that can be modeled combinatorially or algebraically. I will introduce a random combinatorial billiard trajectory depending on some fixed probability p; when p tends to 0, it essentially recovers Thomas Lam's reduced random walk. This random billiard trajectory can also be interpreted as a random growth process on core partitions. The analysis of the random billiard trajectory relies on new finite Markov chains called stoned exclusion processes, which are variants of certain interacting particle systems. These processes have remarkable stationary distributions determined by well-studied polynomials such as ASEP polynomials, inhomogeneous TASEP polynomials, and open boundary ASEP polynomials; in many cases, it was previously not known how to construct Markov chains with these stationary distributions.