Solve the below problem:
Q: College professors who are not yet tenured must frequently play an intricate game against their students. They can decide to make their courses extremely easy, at the risk of boring the better-prepared students; they can aim their courses at the "average student," running the risk of disenchanting the students at both ends of the spectrum; or they can make their courses extremely challenging, thus leaving the lessprepared students "in the dust," so to speak. The ratings they receive for each strategy depend on the number of students in each of the three categories: less-prepared, average, wellprepared. Professor Ogre spent years analyzing the feedback from students in the various categories, and came up with the following payoff matrix. (The payoffs are the scores awarded in the student evaluation on a scale of -2 (highly negative) to +2 (highly positive).)
Student Less Prepared Average Student Well Prepared Easy 2 0 -2 Level Average -1 2 0 Difficult -2 1 2
(a) If half the students in Prof. Ogre's classes are average, a quarter are less prepared, and a quarter are well prepared, what would his average approval rating be if he varied the level of his presentation in such a way as to use each level equally often?
(b) If 50% of the students in Prof. Ogre's classes are average, 30% are less prepared, and 20% are well prepared, at what level should Prof. Ogre present his classes?
(c) What kind of student would be least impressed with Prof. Ogre were he to vary the level of his presentation in such a way as to use each level equally often?