Counts in a two-set Venn diagram, a conditional probability, a test for independence and two selections without replacement

A survey of a class of \(16\) students found that \(12\) of them have art lessons, \(8\) have music lessons and exactly \(1\) has neither. The Venn diagram below shows the two groups, where \(p\), \(q\), \(r\) and \(s\) are numbers of students.

1. 1. Write down the value of \(s\).
   2. Find the value of \(q\).
   3. Write down the value of \(p\) and the value of \(r\).[5]
2. 1. One student from the class is chosen at random. The student has music lessons. Write down the probability that this student also has art lessons.
   2. Hence show that having art lessons and having music lessons are not independent events.[4]
3. Two different students are chosen at random from the class, one after the other. Find the probability that the first has music lessons but not art lessons and the second has art lessons but not music lessons.[4]