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).