At a university in Aberystwyth, students can take courses with the science, engineering, business, or arts faculties. By looking at course enrolment data for the students in the institute, it turns out that:

50% take science courses;

of those students who take science courses, 80% also take engineering courses.

(a) Let S denote the event 'takes science courses', and let E denote the event 'takes engineering courses'. Write the information given in the two bullet points in symbolic form.

(b) Calculate the probability that a randomly chosen student in this university takes both science and engineering courses.

(c) Additional information is now given that 20% of the students in this university do not take any engineering courses.

Calculate the probability that a randomly chosen student in this university who takes engineering courses also takes science courses.

(d) What percentage of students in this university take either science or engineering courses (or both)?

(e) For the students in this university, are the events of taking science courses and taking engineering courses independent? Justify your answer.