Multimodality of the Markov binomial distribution

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 $n$.