Consider the following hypothetical situation at a hypothetical college. Thirty percent of all math majors become computer science majors after one year. Another 60% become physics majors after one year. After a year, 70% of the physics majors become math majors and 10% of the physics majors become computer science majors. In contrast to the other departments, computer science students are usually very happy: only 20% of them become math majors and 20% become physics majors after a year.

(a) Draw a graph that describes the situation.

(b) Give the corresponding adjacency matrix. Notice that it is a doubly stochastic matrix.

(c) Consider three students. If each student is majoring in one of these three areas, indicate the student's probable major after 2 and 4 years. For example, if a student is majoring in Math, what is the probability that he ended up majoring in CS or Phys in 2 and 4 years. Calculate the same probability of a student majoring in Physics or CS.