Problems

1. (10 points) Suppose that among the 151 students registered in CS 361, there are 128 students that have taken both a calculus and a linear algebra class in the past, and there are 2 students that have taken neither.
   1. How many students have taken at least one of those two math classes in the past?
   2. Now suppose furthermore that the number of students that have not taken linear algebra is 4 times the number of students that have not taken calculus. How many students have taken a linear algebra class in the past?
2. (10 points) Textbook problem 3.19. Express your answers as products of fractions or using choose notation. There's no need to compute numerical answers.
3. (10 points) Textbook problem 3.27. Note that each part asks for a single probability. Express your answers using choose and/or summation notation. There's no need to compute numerical answers.
4. (10 points) Suppose that a student is registered this semester in Math 225 (course explorer) and CS 361 (course explorer). In this problem, you will think about the student's registration in combinations of sections of these two classes. Assume that the student's other classes do not conflict with any sections of Math 225 or CS 361.
   1. Write down the sample space of all valid non-conflicting registrations in Math 225 and CS 361. Use the following notation: \( (P1, AL1, ADA) \) is the outcome that means the student is in Math 225 P1 and CS 361 AL1 and ADA.
   2. Now assume that the outcomes you listed in part (a) are equally probable. Let \( E_{P1} \) be the event that the student is registered in Math 225 P1, and so on for other sections.
   1. Are \( E_{P1} \) and \( E_{AL1} \) independent? Justify your answer with calculations.
   2. Are \( E_{P1} \) and \( E_{ADE} \) independent? Justify your answer with calculations.
   3. Are \( E_{S1} \) and \( E_{ADA} \) independent? Justify your answer with calculations.
5. (10 points) Go to this grade disparity data visualization and look up ECE 120: Introduction to Computing. This visualization contains grade data from several semesters of ECE 120. Consider a randomly selected former ECE 120 student whose data is represented in this data set. Let \( A \) be the event that the student received some type of A grade. Let \( V\) be the event that the student had David Varodayan as ECE 120 instructor.
   1. Before doing the calculations below, determine whether \( P(V|A) \) is going to be smaller or larger than \( P(V) \). In one sentence, explain in words why you think so.
   2. Calculate \( P(V) \).
   3. Calculate \( P(V|A) \).
   4. What assumption about the data (that is not actually true) did you have to make to perform the calculations above? Hint: What happened if a student retook ECE 120?