The Math Explorer’s Club is an NSF supported project aiming at
developing materials and activites to give middle school and high
school students an experience of more advanced topics in mathematics.

In this activity, we introduce and develop the notion of Markov chains,
consolidating the student’s grasp of (simple)

probability theory
and
introducing two important

discrete
mathematics concepts: the
relationship between

graphs
and

matrices,
and

recurrence relations.

The
material developed here is not difficult and with graphing calculators,
the computations are not hard. A basic understanding of probability
theory is assumed though. The reader might want to consider having a
look at the

Probability math explorer’s write-up, for example.

We first
introduce the example of a mouse in a maze and develop the idea of a

transition
probability. After playing with this toy model, the next
section introduces the framework of Markov chains and their matrices
showing how it makes it easier to deal with problems like that of the
mouse in great generality. We then investigate the model and then
provide a few applications to practice the learned skills.