Subject choice survey in Form 5
A survey of the 40 students in a Form 5 class asked which of Art (A), Music (M) and Drama (D) they take. The results recorded by the class secretary were:
- n(A) = 20, n(M) = 18, n(D) = 15
- n(A ∩ M) = 8, n(A ∩ D) = 6, n(M ∩ D) = 5
- n(A ∩ M ∩ D) = 3
School rules state that every student must take at least one of these three subjects. The deputy principal wants the figures checked before they are entered into the timetable system, because a subject that is under-subscribed may be withdrawn next year.
1. Using a Venn diagram, find how many students take none of the three subjects. What should the timetable office conclude from your answer?
Read the worked answer
Work outwards from the centre, writing n(A ∩ M ∩ D) = 3 in the middle region.
- A and M only: 8 − 3 = 5
- A and D only: 6 − 3 = 3
- M and D only: 5 − 3 = 2
- A only: 20 − (5 + 3 + 3) = 9
- M only: 18 − (5 + 2 + 3) = 8
- D only: 15 − (3 + 2 + 3) = 7
Total taking at least one subject:
9 + 8 + 7 + 5 + 3 + 2 + 3 = 37.
So the number taking none is 40 − 37 = 3.
Conclusion: the rule that everyone takes at least one of the three subjects is being broken by 3 students, or the secretary's figures contain an error. Either way the data cannot go into the timetable system unchanged: the office must identify the 3 students (or recheck the survey) before the enrolment figures are used to decide whether a subject is under-subscribed.
2. A teacher claims that 'more students take exactly two of these subjects than take exactly one'. Test this claim using the Venn diagram regions.
Read the worked answer
Exactly one subject:
9 (A only) + 8 (M only) + 7 (D only) = 24 students.
Exactly two subjects:
5 (A and M only) + 3 (A and D only) + 2 (M and D only) = 10 students.
The three students in the middle take all three subjects, so they are not counted in either group.
Since 24 > 10, the claim is false: twice as many students (24 against 10) take exactly one of the subjects. The likely reason for the teacher's error is adding the given overlaps 8, 6 and 5, which gives 19 and includes the three students who take all three subjects in every overlap.
3. One student is chosen at random from the 40. Find the probability that the student takes Drama but not Art. Give the answer as a fraction in its simplest form.
Read the worked answer
Drama but not Art covers two regions of the Venn diagram:
- Drama only: 7
- Music and Drama only: 2
Total = 7 + 2 = 9 students.
The region 'Art and Drama only' (3 students) and the middle region (3 students) are excluded because those students do take Art. Check against n(D) = 15: 7 + 2 + 3 + 3 = 15.
Probability = 9/40.
9 and 40 have no common factor other than 1, so the answer is 9/40 (0.225, or 22.5%).