This semester the class has a total of 140 students, two regular classes and one exercise class per week. Based on what was learned during the first week of classes, the teacher estimates that each student has a probability qc of going to normal classes and a probability qc of going to exercise classes (0 < qa < qc < 1). Assuming that the students are independent of each other, the number of students who go to class on any given day distributes Binomial (140, qc) if it is a normal class and Binomial (140, qa) if it is an exercise class. Assume that these distributions are independent of each other and independent for different days.

The teacher reminds that the probabilities of going to normal classes are pretty good for a normal week, but these can vary if it is a week of tests or if it is the university week. Therefore, the teacher models the probability of going to normal classes for any student as a discrete random variable Q. This variable takes value pn if it is a normal week, pp if it is a test week and pu if it is a university week (0 < pn, pp, pu < 1). She considers that with probability 0.6 the week is normal, 0.3 is the testing week and 0.1 is the university week. She again, she considers that the students are independent of each other and, therefore, the number of students that go to any normal class is given by Binomial (140, Q).

(c) Find the probability that 90 students will arrive in any regular class in a week that is NOT college week (that is, it can be regular or testing).

(d) Find the expected number of students arriving in a typical class in any given week.