Mathematics • Year 10 • Unit 4 • Lesson 17
Venn Diagrams and Two-Way Tables in the Real World
Apply Lesson 17's Venn diagrams, two-way tables and the addition rule to school surveys, sports streaming data, public health snapshots and an Australian census-style scenario.
1. Word problems
Use Venn diagrams or two-way tables as appropriate. Show working.
1.1 — Year 10 elective survey. A survey of 100 Year 10 students: 60 chose Visual Arts, 45 chose Music, and 25 chose both.
(a) Build a Venn diagram (or sketch it) and fill in all four regions.
(b) Find P(Visual Arts or Music) and P(neither). 3 marks
1.2 — Streaming services. Of 200 households surveyed: 130 subscribe to Netflix, 90 subscribe to Stan, 50 subscribe to both.
(a) Build a two-way table (rows = Netflix Y/N, columns = Stan Y/N).
(b) Find P(at least one of the two services) and P(neither service). 3 marks
1.3 — Standard deck of cards. A card is drawn at random from a standard 52-card deck. Let H = "heart", K = "king", B = "black card".
(a) Are H and B mutually exclusive? Justify.
(b) Are H and K complementary? Justify.
(c) Find P(H or K) using the addition rule. 3 marks
1.4 — Public health quick-survey. A clinic surveys 400 adults: 240 are vaccinated against flu, 180 are vaccinated against COVID, 130 are vaccinated against both.
(a) Build a Venn diagram and a two-way table for "Flu Y/N" × "COVID Y/N".
(b) Find P(at least one vaccine) and P(no vaccines).
(c) Use the Lesson 17 Exam Tip to verify that your row and column totals match. 3 marks
1.5 — School sports register. 120 Year 10 students. The PE teacher records: 70 play a winter sport, 60 play a summer sport. Every student plays at least one sport (no "neither" region).
(a) How many students play both a winter and a summer sport? (Use n(W or S) = 120 with the addition rule.)
(b) Find P(plays exactly one sport, not both). 3 marks
2. Explain your thinking
Communication question. Use full sentences. 4 marks
2.1 A friend is studying for the Yr 10 exam and says: "Mutually exclusive and complementary mean the same thing — they're both events that can't happen at the same time." Write a four-sentence reply that (i) corrects the misconception using the Lesson 17 fix, (ii) gives one example of two events that are mutually exclusive but not complementary (use a die or a card deck), (iii) gives one example of complementary events from the same sample space, and (iv) finishes with a one-sentence rule a Year 10 student can use to tell the two apart.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Visual Arts / Music
VA only = 35, both = 25, Music only = 20, neither = 100 − 80 = 20.
(b) P(VA or M) = (35+25+20)/100 = 80/100 = 4/5. P(neither) = 20/100 = 1/5.
1.2 — Netflix / Stan
Netflix only = 80, both = 50, Stan only = 40, neither = 200 − 170 = 30.
Two-way table:
| Stan Y | Stan N | Total
Net Y | 50 | 80 | 130
Net N | 40 | 30 | 70
Tot | 90 | 110 | 200 ✓
(b) P(at least one) = 170/200 = 17/20 = 0.85. P(neither) = 30/200 = 3/20 = 0.15.
1.3 — Cards
(a) H and B are mutually exclusive: hearts are red, so a card cannot be a heart AND black.
(b) H and K are not complementary. They can overlap (king of hearts) AND many cards are neither a heart nor a king. To be complementary they would have to be mutually exclusive and cover the whole deck.
(c) P(H or K) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13.
1.4 — Vaccination survey
Flu only = 240 − 130 = 110. Both = 130. COVID only = 180 − 130 = 50. Neither = 400 − 290 = 110.
Two-way table:
| COVID Y | COVID N | Total
Flu Y | 130 | 110 | 240
Flu N | 50 | 110 | 160
Tot | 180 | 220 | 400 ✓
(b) P(at least one) = 290/400 = 29/40 = 0.725. P(no vaccines) = 110/400 = 11/40 = 0.275.
(c) Row totals 240+160 = 400 ✓; column totals 180+220 = 400 ✓.
1.5 — Winter and summer sports
(a) n(W or S) = n(W) + n(S) − n(both). 120 = 70 + 60 − n(both), so n(both) = 10.
(b) Exactly one sport: (70−10) + (60−10) = 60 + 50 = 110. P(exactly one) = 110/120 = 11/12.
2.1 — Explain your thinking (sample response)
The friend has mixed up two different ideas. Mutually exclusive means the two events cannot both occur — but neither one is guaranteed to occur. Complementary is the stronger version: mutually exclusive AND together they cover the whole sample space (so exactly one MUST occur). For example, on a fair die, "even" and "rolling a 3" are mutually exclusive (3 is odd, so no overlap) but they are not complementary, because rolling a 1 or a 5 is neither. By contrast, "even" and "odd" are complementary on a die — together they cover every outcome. Rule of thumb: if P(A) + P(B) = 1 and they cannot both occur, they are complementary; otherwise mutually exclusive is the most you can say.
Marking: 1 mark per part (i)-(iv).