Question

Suppose a math professor collects data on the probability that students attending a given class meeting will attend the next one. He finds that 95% of students who attended a given class meeting will attend the following class meeting and that 25% of students who do not attend a given class meeting will not attend the next one. Build a discrete dynamical system model using linear algebra. Be sure to state your transition matrix explicitly. What percentage of students does your model predict will be attending class meetings by the end of the semester (in the long run)?

Answer 1

We would be using a markov model here to represent the given situation. The two states for the model here are given as:

• Student attending a class
• Student not attending the class

This can be represented in the form of a probability transition matrix for the given probabilities here as:

Let the long run probabilities here for attending the class and not attending the class be X and Y respectively.

Then, from first column, we have:
X = 0.95X + 0.75Y
0.05X = 0.75Y
X = 15Y

Also, we know here that:
X + Y = 1
16Y = 1
Y = 1/16

Therefore in long run, 15/16 = 93.75% of the students would be attending the class meeting here.