Mathematics • Year 8 • Unit 4 • Lesson 17
Venn Diagrams — Mixed Challenge
Bring everything from Lesson 17 together: filling diagrams from raw counts, the addition rule with probabilities, mutually exclusive events, and a stretch three-set diagram. Six mixed problems, one "find the mistake", and one open-ended design challenge.
1. Mixed problems — choose the right move
Each question uses a different idea from Lesson 17. Show your working. 3 marks each
1.1 n(ξ) = 60, n(A) = 28, n(B) = 22, n(A ∩ B) = 12. Find n(A only), n(B only), and n(neither). Verify your four regions sum to 60.
1.2 Use the addition rule with P(A) = 0.55, P(B) = 0.40, and P(A ∩ B) = 0.25 to find P(A ∪ B). Then state P(neither).
1.3 A class of 40 students: 25 like maths, 18 like science, 10 like neither. (a) How many like at least one subject? (b) Use the addition rule to find how many like both.
1.4 A card is drawn from a standard 52-card deck. Let A = "card is a heart" (13 hearts) and B = "card is a king" (4 kings). Find P(A), P(B), P(A ∩ B), and P(A ∪ B). Are A and B mutually exclusive? Justify.
1.5 For events with P(A) = 0.5, P(B) = 0.3 and P(A ∪ B) = 0.7, rearrange the addition rule to find P(A ∩ B). Then state whether A and B are mutually exclusive.
1.6 A school of 100 surveyed walkers (W) and bus riders (B): 45 walk to school, 30 take the bus, 10 do both (walk part-way then bus), and the rest are dropped off by car. (a) Find how many are dropped off by car. (b) Find P(walks AND takes the bus). (c) Find P(walks OR takes the bus).
2. Find the mistake
A student attempted this Venn-diagram problem. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Problem: n(ξ) = 50, n(A) = 24, n(B) = 18, n(A ∩ B) = 8. Find n(neither).
Line 1: Place intersection: n(A ∩ B) = 8 in overlap.
Line 2: A only = 24, B only = 18.
Line 3: Neither = 50 − 24 − 8 − 18 = 0.
Line 4: So no one is in "neither".
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full and state n(neither).
Stuck? "A only" is NOT the same as n(A). The student needs to subtract n(A ∩ B) from n(A) before writing the "only" region.3. Open-ended challenge — design your own data survey
This question has many valid answers. 4 marks
3.1 Your job: invent a realistic two-set Venn-diagram scenario for a school of 200 students. Choose any two activities/preferences (e.g., "plays a musical instrument" and "speaks a second language at home").
Your answer must include:
(i) State the universal set (n(ξ) = 200) and your two events A and B.
(ii) Invent realistic counts for n(A), n(B), and n(A ∩ B). The numbers must satisfy n(A ∩ B) ≤ min(n(A), n(B)) AND n(A) + n(B) − n(A ∩ B) ≤ 200.
(iii) Fill all four regions of the Venn diagram and verify they sum to 200.
(iv) Calculate P(A ∪ B) and P(neither). Then check the addition rule numerically.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Four-region fill
A only = 28 − 12 = 16; A ∩ B = 12; B only = 22 − 12 = 10; Neither = 60 − 16 − 12 − 10 = 22. Check: 16 + 12 + 10 + 22 = 60 ✓.
1.2 — Addition rule with probabilities
P(A ∪ B) = 0.55 + 0.40 − 0.25 = 0.70. P(neither) = 1 − 0.70 = 0.30.
1.3 — Maths and science
(a) At least one = 40 − 10 = 30.
(b) 30 = 25 + 18 − n(both), so n(both) = 43 − 30 = 13.
1.4 — Cards: heart and king
P(A) = 13/52 = 1/4. P(B) = 4/52 = 1/13. P(A ∩ B) = 1/52 (king of hearts). P(A ∪ B) = 13/52 + 4/52 − 1/52 = 16/52 = 4/13. NOT mutually exclusive — the king of hearts is in both events, so P(A ∩ B) ≠ 0.
1.5 — Rearrange the addition rule
P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.5 + 0.3 − 0.7 = 0.1. Not mutually exclusive (P(A ∩ B) = 0.1, not 0).
1.6 — Walking and bus
W only = 45 − 10 = 35. B only = 30 − 10 = 20. Both = 10. (a) Car (neither) = 100 − 35 − 10 − 20 = 35.
(b) P(walks AND bus) = 10 / 100 = 1/10.
(c) P(walks OR bus) = 65 / 100 = 13/20 (or use addition rule: 45/100 + 30/100 − 10/100 = 65/100).
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student wrote n(A) = 24 in the "A only" region without subtracting the intersection. A only must be n(A) − n(A ∩ B) = 24 − 8 = 16, not 24. Same fix for B only: 18 − 8 = 10. Otherwise the intersection is counted twice and "neither" comes out as 0 wrongly.
(c) Corrected working:
A only = 24 − 8 = 16. B only = 18 − 8 = 10. A ∩ B = 8.
Neither = 50 − 16 − 8 − 10 = 16. Check: 16 + 8 + 10 + 16 = 50 ✓.
3 — Open-ended design (sample solution)
(i) Universal set: n(ξ) = 200 students at the school. A = plays a musical instrument; B = speaks a second language at home.
(ii) Invented counts: n(A) = 60, n(B) = 80, n(A ∩ B) = 25. Constraint check: 25 ≤ min(60, 80) = 60 ✓; 60 + 80 − 25 = 115 ≤ 200 ✓.
(iii) Regions:
A only = 60 − 25 = 35; A ∩ B = 25; B only = 80 − 25 = 55; Neither = 200 − 35 − 25 − 55 = 85. Check: 35 + 25 + 55 + 85 = 200 ✓.
(iv) Probabilities:
P(A ∪ B) = (35 + 25 + 55) / 200 = 115 / 200 = 23/40.
P(neither) = 85 / 200 = 17/40.
Addition-rule check: P(A) + P(B) − P(A ∩ B) = 60/200 + 80/200 − 25/200 = 115/200 = 23/40 ✓ — matches the direct count.
Marking: 1 mark for clear universal set and events; 1 mark for invented counts that satisfy the inequalities; 1 mark for filling all four regions correctly; 1 mark for both probabilities AND addition-rule verification.