In this homework, whenever you are applying a rule for independence of events you must:

- specify which events are supposed to be independent

**(10 points)**Suppose that among the 220 students registered in CS 361, there are 185 students that have taken both a calculus and a linear algebra class in the past, and there are 5 students that have taken neither.- How many students have taken at least one of those two math classes in the past?
- 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?

**(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.**(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.**(10 points)**Suppose that a student is registered in a previous 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.- Write down the sample space of all valid non-conflicting registrations in Math 225 and CS 361. Use the following notation: \( (S1, AL1, ADA) \) is the outcome that means the student is in Math 225 S1 and CS 361 AL1 and ADA.
- Now assume that the outcomes you listed in part (a) are equally probable. Let \( E_{S1} \) be the event that the student is registered in Math 225 S1, and so on for other sections.
- Are \( E_{S1} \) and \( E_{AL1} \) independent? Justify your answer with calculations.
- Are \( E_{S1} \) and \( E_{ADE} \) independent? Justify your answer with calculations.

**(10 points)**Textbook problem 3.44.**(Extra credit: 3 points)**Let A, B and C are events in a sample space while A and B are disjoint events. We know \( P(A) = 2P(B), P(C|A)=2/7, P(C|B)= 4/7. \) What is \( P(C|(A\cup B))? \)