SWBAT use a Markov chain to see how a population transitions over time.

A Markov chain combines matrices and probability in order to analyze the Purrrrrfect Cat Toy's market share.

10 minutes

A Markov chain is a mathematical model that can be used to show how a population changes over time based on its current state. For example, if users of Brand A toothpaste stick with Brand A 63% of the time and those who don’t use Brand A switch to Brand A 45% of the time. A Markov chain will help you find out what will happen to the percentage of Brand A users over many time cycles. For example, a Markov chain can help you predict what percentage of customers will use Brand A in three years if 30% of a population currently use it.

These YouTube videos (also embedded below) are a good crash course on Markov chains if you are not familiar with them. They really are a nice extension of matrices and a good reminder about probability.

For a warm up to today's lesson, I give students the top portion of the worksheet so that they can familiarize themselves with some probability concepts. Particularly, I want the class to remember that when two independent events happen consecutively we multiply their probabilities. And, we add the probabilities if we know the probability of certain subsets of a sample space.

**Instructional Note**: If students don't have a strong probability background, you might want to take a day to give them some experience with "and" and "or" problems.

30 minutes

Today’s lesson is one where students will require slightly more direct instruction than normal – there has to be a certain structure to the matrices that we use, so I guide my students to get to that point. However, I still expect them to have a conceptual understanding of what is going on and specifically how the matrix multiplication relates to probability.

The format for today’s lesson is more of a guided class discussion. We will go through each problem from the worksheet together and then move on to the next. For each problem I will give them time to brainstorm with their table groups before we discuss – I am still not going to just give them the answer.

For question a), you might want to start by walking through a diagram like this so that students understand what exactly is going on with the transitions between the three states ( buying a toy from the Purrrrrfect Cat Toy Company, buying a toy from another cat toy company, and not buying a cat toy). Once you set up the diagram with the class, you can allow them to try to make the transition matrix on their own and see what they come up with.

For question c), students are to find *T*^{2} and then interpret its meaning. Although I would suggest that you get the answer using a graphing calculator for maximum efficiency, I find it very helpful to think about where each entry comes from in order to decode the meaning of *T*^{2}. In the video below I give some helpful hints to connect the probability review from the Launch to this new learning. Also, here is the image from the video.

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For the remaining questions, we are exploring how a specific market will change based on its starting state. Students will likely realize that we need a new matrix to represent the starting percentages of Denver. Deciding on the dimensions may be tricky – I have students experiment until they realize that it must be 3 by 1.

I always find it fascinating the market will stabilize in the long run! As *T* is raised to really high powers, the three categories will stop changing and will each reach their own “limit.” It doesn’t matter what Denver starts out at, after many years the distribution will always be the same! You can have students play around with the starting percentages of Denver for them to come to this conclusion.

15 minutes

After going through the notes worksheet, students might be a little overwhelmed. It is a good idea to wrap up with a few big ideas about a Markov chain. I have students share their thoughts on what we did in class today. Here are a few things that I try to get out of them before we end the lesson:

- The state of the population will change after each cycle because people are constantly transitioning and changing groups.
- The matrix multiplication accounts for those transitions since there are three products that combine to make each sum.
- We can raise the transition matrix to higher powers to find many transitions at once.
- Eventually the states will stabilize and they will have their own “limit.”
- Like with the communication matrices, using a matrix simplifies the process of finding many individual probabilities. It is an efficient method.

Finally, here is a homework assignment that I will give students to give them some more practice with these concepts.