Mathematics • Year 8 • Unit 4 • Lesson 12
Two-Way Tables — Mixed Challenge
Pull together cell reading, completing tables, joint vs conditional probability, and writing comparative claims. Six mixed problems, one "find the mistake", and one open-ended challenge.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 12. Show your working. 3 marks each
1.1 80 students were asked if they own a bike (Yes/No) and if they ride to school (Yes/No). Own + Rides = 28, Own + Doesn't Ride = 22, Doesn't Own + Rides = 0, Doesn't Own + Doesn't Ride = 30. Complete the table with all row and column totals. Then state the grand total.
1.2 Using the bike data above: (a) calculate P(Owns a bike). (b) Calculate P(Rides | Owns a bike). (c) Calculate P(Rides | Doesn't own a bike). What does the comparison tell you?
1.3 A two-way table has rows Male/Female and columns Glasses/No Glasses. Grand total = 300. Male + Glasses = 60, Female + Glasses = 90, Female + No Glasses = 70. Find Male + No Glasses, both row totals, and both column totals.
1.4 Of 50 surveyed teenagers, 20 own a phone AND have a TikTok account, 8 own a phone but no TikTok, 0 don't own a phone but have TikTok, 22 own neither. Construct the two-way table. Comment on which combination is impossible in this data and why.
1.5 A 2 × 2 table has row totals 60 and 40 (grand total 100). The top-left cell is 25. Without any other information, can you fill in all the cells? Explain how (or why not). If yes, find them all.
1.6 A surveyor claims: "Owning a pet is linked to being happier." Their two-way table for 200 people shows Has Pet + Happy = 80, Has Pet + Not Happy = 20, No Pet + Happy = 60, No Pet + Not Happy = 40. Calculate P(Happy | Has Pet) and P(Happy | No Pet). Does the data support the claim? Quote the numbers.
2. Find the mistake
Another student calculated a conditional probability from a two-way table. Exactly one line below contains a mistake. Spot it, explain why it is wrong, then re-do the working correctly. 3 marks
Problem: Of 120 students surveyed about sport and music: Sport + Music = 35, Sport + No Music = 45, No Sport + Music = 25, No Sport + No Music = 15. Find P(Music | Sport).
Line 1: Sport row total = 35 + 45 = 80.
Line 2: Music column total = 35 + 25 = 60.
Line 3: P(Music | Sport) means "given Sport, what is the probability of Music?"
Line 4: P(Music | Sport) = (Sport AND Music) ÷ (Music column total) = 35 ÷ 60 ≈ 58%.
Line 5: Answer: 58%.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write the corrected working and the correct answer.
Stuck? For P(B | A), the denominator is the row total of A — the "given" variable. The student used the wrong margin.3. Open-ended challenge — design your own survey table
This question has many valid answers. 4 marks
3.1 Your job: design a 2 × 2 survey table where the data shows a clear link between two variables.
(i) Choose ONE topic for your survey (e.g. study hours vs grades, breakfast vs alertness, social media vs sleep) and write the row variable and column variable.
(ii) Decide on a grand total between 100 and 300, and invent four cell counts so that the conditional probabilities show a clear link (e.g. one row has much higher % for one column than the other row).
(iii) Draw the complete table including row totals, column totals, and grand total. Check that row totals add to grand total AND column totals add to grand total.
(iv) Calculate the two conditional probabilities that prove the link (give both as percentages).
(v) Write a one-sentence headline a journalist could use, based on your data (e.g. "Daily breakfast linked to 30% higher alertness in students").
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Bike table
Own row: 28, 22, total 50. Doesn't Own row: 0, 30, total 30. Rides column total = 28 + 0 = 28. Doesn't Ride column total = 22 + 30 = 52. Grand total = 80. Check: 28 + 52 = 80 ✓; 50 + 30 = 80 ✓.
1.2 — Bike conditionals
(a) P(Owns bike) = 50 ÷ 80 = 62.5%.
(b) P(Rides | Owns) = 28 ÷ 50 = 56%.
(c) P(Rides | Doesn't Own) = 0 ÷ 30 = 0%.
Only bike owners ride to school (no one rides a bike they don't have!). Owning a bike is a strict precondition for riding.
1.3 — Glasses table
Male + No Glasses = total − known cells = 300 − 60 − 90 − 70 = 80. Male row total = 60 + 80 = 140. Female row total = 90 + 70 = 160. Glasses column = 60 + 90 = 150. No Glasses column = 80 + 70 = 150. Check: 140 + 160 = 300 ✓.
1.4 — Phone and TikTok
Table: Phone row (20, 8, total 28). No Phone row (0, 22, total 22). Columns: TikTok = 20, No TikTok = 30, grand total = 50. The combination "No Phone + TikTok" = 0 — impossible in this group because you can't have a TikTok account that you use without owning a phone (or another device — in the survey assumptions, accessing TikTok requires owning a phone).
1.5 — 2×2 with row totals only
Yes — the top-left cell plus row totals plus grand total determines everything. Top-right = 60 − 25 = 35. Bottom-left = column total − 25; first we need the columns. Top-left + bottom-left = column 1 total, and column 1 + column 2 totals = 100. But we don't know column 1 directly... actually we DO: top-left + bottom-left = col 1; bottom-left = 40 − bottom-right; and col 1 + col 2 = 100. With one row total (40) and top-left = 25, the bottom row is unknown until we use one more clue. Two cells from different rows AND different columns plus the totals are needed — one cell and the row totals is NOT enough. So the answer is: NO, the table is not fully determined without one more piece of information.
1.6 — Pets and happiness
Pet owners row total = 80 + 20 = 100. No-pet row total = 60 + 40 = 100.
P(Happy | Has Pet) = 80 ÷ 100 = 80%.
P(Happy | No Pet) = 60 ÷ 100 = 60%.
The data supports the claim — pet owners are happier (80%) than non-owners (60%), a 20 percentage-point gap. Caution: correlation, not necessarily causation — happy people might be more likely to take on pets, rather than pets making people happier.
2 — Find the mistake
(a) The mistake is on Line 4.
(b) For P(Music | Sport), the denominator must be the Sport row total (the "given" condition), not the Music column total. The student used the wrong margin.
(c) Corrected: P(Music | Sport) = (Sport AND Music) ÷ (Sport row total) = 35 ÷ 80 = 0.4375 = 43.75%. Answer ≈ 44%, not 58%. (Rule: the "given" variable goes in the denominator.)
3 — Open-ended challenge (sample solution)
(i) Topic: Daily breakfast vs morning alertness in Year 8 students. Rows: Eats breakfast / No breakfast. Columns: Alert / Sleepy.
(ii) Cells: Eats breakfast + Alert = 80, Eats breakfast + Sleepy = 20, No breakfast + Alert = 30, No breakfast + Sleepy = 70. Grand total = 200.
(iii) Table check: Eats breakfast row total = 100. No breakfast row total = 100. Alert col total = 110. Sleepy col total = 90. 100 + 100 = 200 ✓; 110 + 90 = 200 ✓.
(iv) Conditionals: P(Alert | Eats breakfast) = 80 ÷ 100 = 80%. P(Alert | No breakfast) = 30 ÷ 100 = 30%.
(v) Headline: "Year 8 students who eat breakfast are nearly three times more likely to feel alert in the morning (80% vs 30%)."
Marking: 1 mark for valid topic and row/column variables. 1 mark for a complete, consistent table (all checks pass). 1 mark for two correct conditional probabilities expressed as percentages. 1 mark for a clear, accurate one-sentence headline that uses the numbers.