For an article on machine learning, the New York Times implemented a Rock-Paper-Scissors playing algorithm that utilizes a 4th-order Markov chain (The next state depends on the current state and the previous 3.) Play with it in "Veteran" mode to see how powerful a fairly simple probabilistic model can be.
Things to think about: How would you represent this model as a state diagram? As a formal grammar? How would you parameterize such a model?