  We study the shape of the probability mass function of the Markov binomial
distribution, and give necessary and sufficient conditions for the probability
mass function to be unimodal, bimodal or trimodal. These are useful to analyze
the double-peaking results from a PDE reactive transport model from the
engineering literature. Moreover, we give a closed form expression for the
variance of the Markov binomial distribution, and expressions for the mean and
the variance conditioned on the state at time $n$.
