In a survey of students in a business statistics class, students were asked what grade they expected in the course, and also whether they worked additional problems beyond those assigned by the instructor. The table gives proportions of students in each of eight joint classifications.

(a) Find the probability that a randomly chosen student from this class worked additional problems.

(b) Find the probability that a randomly chosen student from this class expects an A.

(c) Find the probability that a randomly chosen student who worked additional problems expects an A.

(d) Find the probability that a randomly chosen student who expects an A worked addi- tional problems.

(e) Find the probability that a randomly chosen student who worked additional problems expects a grade below B.

(f) Are working additional problems and expected grade statistically independent?