A professor continually gives exams to her students. She can give three possi- ble types of exams, and her class is graded as either having done well or badly. Let pi denote the probability that the class does well on a type i exam, and suppose that p1 = 0.3, p2 = 0.6, and p3 = 0.9. If the class does well on an exam, then the next exam is equally likely to be any of the three types. If the class does badly, then the next exam is always type 1. What proportion of exams are type i, i = 1, 2, 3?