Mathematics • Year 8 • Unit 4 • Lesson 12
Two-Way Tables in the Real World
Apply two-way tables to real surveys: health and screen time, handedness, transport choices, school subjects, and market research. Each problem demands both calculation and interpretation.
1. Word problems
Each problem uses ideas from Lesson 12 — completing tables, joint and conditional probability, and interpretation. Show your working.
1.1 — Screen time and sleep. 150 students were surveyed: High Screen + Good Sleep = 12, High Screen + Poor Sleep = 48, Low Screen + Good Sleep = 54, Low Screen + Poor Sleep = 36.
(a) Draw the two-way table including all row and column totals.
(b) Calculate P(High Screen AND Poor Sleep).
(c) Calculate P(Poor Sleep | High Screen) AND P(Poor Sleep | Low Screen). Compare the two values in one sentence. 4 marks
1.2 — Handedness and sport. 200 students were surveyed about which hand they write with and which hand they prefer for sport:
Right-handed + Right sport = 140 Right-handed + Left sport = 20
Left-handed + Right sport = 10 Left-handed + Left sport = 30
(a) Find both row totals (left-handed total, right-handed total).
(b) Calculate P(prefers Right sport | Right-handed).
(c) Calculate P(prefers Right sport | Left-handed).
(d) Does handedness seem linked to sport-hand preference? Justify in one sentence. 4 marks
1.3 — Transport survey. A school polls 240 students about how they get to school and whether they live within 2 km.
Lives Near + Walks = 60 Lives Near + Catches Bus = 20
Lives Far + Walks = 15 Lives Far + Catches Bus = 145
(a) What is the grand total in the table? Does it match the survey size?
(b) Calculate P(Walks). (c) Calculate P(Walks | Lives Near). (d) Why are these two probabilities so different? 4 marks
1.4 — Subject choice. 180 Year 8 students were surveyed about whether they enjoy Maths and whether they enjoy Science.
Enjoys Maths + Enjoys Science = 70 Enjoys Maths + Doesn't = 30
Doesn't Maths + Enjoys Science = 50 Doesn't Maths + Doesn't = 30
(a) Find the row totals and column totals.
(b) What percentage of all students enjoy Science?
(c) Of those who enjoy Maths, what percentage also enjoy Science?
(d) Is enjoying Maths linked to enjoying Science? Quote the comparison. 4 marks
1.5 — Market research. A snack company surveyed 400 shoppers about whether they tasted the new biscuit AND whether they would buy it.
Tasted + Would Buy = 144 Tasted + Wouldn't Buy = 56
Didn't Taste + Would Buy = 60 Didn't Taste + Wouldn't Buy = 140
(a) Calculate P(Would Buy | Tasted).
(b) Calculate P(Would Buy | Didn't Taste).
(c) The marketing team claims "tasting the biscuit makes shoppers 2.4 times more likely to buy". Use your two percentages to check whether that claim is correct (show the ratio). 3 marks
2. Explain your thinking
This question is about communication, not just numbers. Use full sentences. 4 marks
2.1 A student is calculating "the probability that a student plays sport". They find the Sport+Music cell = 35 in a table with grand total 120 and write P(Sport) = 35 ÷ 120 = 29%. Explain (i) what mistake they have made, (ii) what they SHOULD have used as the numerator (with the correct numerical value), and (iii) write a one-line rule a Year 8 student could memorise that prevents this mistake. Use the words joint frequency and marginal frequency.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Screen and sleep
(a) High Screen row total = 60. Low Screen row total = 90. Good Sleep column = 66. Poor Sleep column = 84. Grand total = 150.
(b) P(High Screen AND Poor Sleep) = 48 ÷ 150 = 0.32 = 32%.
(c) P(Poor Sleep | High Screen) = 48 ÷ 60 = 80%. P(Poor Sleep | Low Screen) = 36 ÷ 90 = 40%. High-screen students are twice as likely to have poor sleep as low-screen students, suggesting a strong link.
1.2 — Handedness and sport
(a) Right-handed row total = 140 + 20 = 160. Left-handed row total = 10 + 30 = 40.
(b) P(Right sport | Right-handed) = 140 ÷ 160 = 87.5%.
(c) P(Right sport | Left-handed) = 10 ÷ 40 = 25%.
(d) Yes — handedness strongly predicts sport-hand preference. Right-handed people overwhelmingly play right-handed (87.5%); left-handed people mostly prefer left-handed sport (only 25% switch to right).
1.3 — Transport survey
(a) Grand total = 60 + 20 + 15 + 145 = 240 ✓ (matches survey size).
(b) P(Walks) = (60 + 15) ÷ 240 = 75 ÷ 240 = 31.25%.
(c) P(Walks | Lives Near) = 60 ÷ (60 + 20) = 60 ÷ 80 = 75%.
(d) The conditional probability (75%) is more than double the unconditional one (31.25%) because nearly all students who walk live within 2 km — knowing a student lives near greatly increases the chance they walk.
1.4 — Subject choice
(a) Enjoys Maths row total = 70 + 30 = 100. Doesn't Maths row total = 50 + 30 = 80. Enjoys Science col = 70 + 50 = 120. Doesn't Science col = 30 + 30 = 60. Grand total = 180.
(b) P(Science) = 120 ÷ 180 ≈ 66.7%.
(c) P(Science | Maths) = 70 ÷ 100 = 70%.
(d) Slight link only — 70% vs 66.7% is a small difference (about 3 percentage points), so enjoying Maths is only weakly associated with enjoying Science here.
1.5 — Snack market research
(a) P(Would Buy | Tasted) = 144 ÷ 200 = 72%.
(b) P(Would Buy | Didn't Taste) = 60 ÷ 200 = 30%.
(c) Ratio = 72 ÷ 30 = 2.4. The marketing claim is correct — tasters are exactly 2.4 times more likely to say they would buy.
2.1 — Explain your thinking (sample response)
The mistake is using a joint frequency (the cell Sport AND Music = 35) when they should have used the marginal frequency for sport (the Sport row total). The cell counts only students who play both sport and music — it leaves out 45 students who play sport but no music. The correct numerator is the Sport row total = 80, so P(Sport) = 80 ÷ 120 ≈ 66.7%, not 29%. A one-line rule: "To find P(category X), use the row or column TOTAL for X — not a cell."
Marking: 1 mark for naming the mistake (cell vs total); 1 mark for the correct numerator (80) and answer (66.7%); 1 mark for using the terms "joint" and "marginal" correctly; 1 mark for a memorable one-line rule.