Mathematics • Year 9 • Unit 4 • Lesson 15
Venn Diagrams in the Real World
Use Venn diagrams and the addition rule on real situations — pet owners, Year 9 language electives, streaming subscriptions, public-health co-occurrence and survey overlap. Then explain why we must subtract the overlap.
1. Word problems
Each problem uses sets, Venn diagrams or the addition rule from Lesson 15. Show your working — a single final answer with no working only earns half marks.
1.1 — Pet owners. A survey of 30 Year 9 students found 20 have a dog, 15 have a cat and 10 have both.
(a) Draw a Venn diagram showing dog only, cat only, both, and neither.
(b) Find P(dog or cat). 4 marks
1.2 — Language electives. In a Year 9 cohort of 80 students, 35 study French, 28 study Japanese and 12 study both.
(a) How many study at least one language?
(b) How many study neither?
(c) Find P(student studies exactly one language). 4 marks
1.3 — Streaming subscriptions. In a survey of 200 families, 130 have Netflix, 90 have Disney+ and 50 have both.
(a) How many have at least one of Netflix or Disney+?
(b) How many have only Netflix?
(c) Find P(a randomly chosen family has neither). 4 marks
1.4 — Public-health co-occurrence. In a regional cohort of 500 adults studied by AIHW, 120 have diabetes, 75 have heart disease and 30 have both.
(a) How many have at least one of the two conditions?
(b) Find P(an adult chosen at random has only diabetes). 4 marks
1.5 — Survey overlap (finding the intersection). A Year 9 wellbeing survey shows that 60% of students do regular exercise (E), 40% get 8+ hours of sleep (S), and 75% do at least one of these two.
(a) Use the addition rule to find P(does both E and S).
(b) Find P(does neither). 4 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate says: "If 60% of Year 9 students play sport and 40% play music, then 100% must play at least one of the two." Using a class of 100 students with sport and music sets that overlap, explain in your own words (i) why adding 60 + 40 directly is wrong, (ii) what the overlap represents, and (iii) how the addition rule fixes the problem. Refer to "double-counting" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pet owners
Dog ∩ cat = 10. Dog only = 20 − 10 = 10. Cat only = 15 − 10 = 5. Inside circles = 10 + 10 + 5 = 25. Neither = 30 − 25 = 5.
(b) P(dog ∪ cat) = 25/30 = 5/6.
1.2 — Language electives
(a) French ∪ Japanese = 35 + 28 − 12 = 51 students.
(b) Neither = 80 − 51 = 29.
(c) French only = 35 − 12 = 23. Japanese only = 28 − 12 = 16. Exactly one = 23 + 16 = 39. P(exactly one) = 39/80 = 0.4875.
1.3 — Streaming subscriptions
(a) Netflix ∪ Disney+ = 130 + 90 − 50 = 170 families.
(b) Netflix only = 130 − 50 = 80.
(c) Neither = 200 − 170 = 30. P(neither) = 30/200 = 3/20 (= 0.15 = 15%).
1.4 — Public-health co-occurrence
(a) Diabetes ∪ heart disease = 120 + 75 − 30 = 165 adults.
(b) Only diabetes = 120 − 30 = 90. P(only diabetes) = 90/500 = 9/50 (= 0.18 = 18%).
1.5 — Survey overlap
(a) Addition rule: P(E ∩ S) = P(E) + P(S) − P(E ∪ S) = 0.60 + 0.40 − 0.75 = 0.25 (25% do both).
(b) P(does neither) = 1 − P(E ∪ S) = 1 − 0.75 = 0.25 (25%).
2.1 — Explain your thinking (sample response)
Imagine 100 Year 9 students where 60 play sport (S) and 40 play music (M). If we just add 60 + 40 = 100, we've assumed every student plays one or the other but never both. In reality some students play both sport and music — that's the overlap of the two sets on a Venn diagram. Those students get counted once in the 60 sport players and once again in the 40 music players, which is exactly the double-counting mistake. The addition rule fixes this by subtracting the overlap once: P(S ∪ M) = P(S) + P(M) − P(S ∩ M). For example, if 20 students play both, then P(S ∪ M) = 60/100 + 40/100 − 20/100 = 80/100 = 0.8, not 1. So 20% of students do neither — exactly what the simple 60 + 40 = 100 calculation missed.
Marking: 1 mark for spotting the overlap; 1 mark for naming "double-counting"; 1 mark for writing the addition rule correctly; 1 mark for using a concrete number example.