Mathematics • Year 8 • Unit 4 • Lesson 17
Venn Diagrams
Build fluency with filling in two-set Venn diagrams, calculating probabilities from each region, and using the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
1. I do — fully worked example
Watch how we fill a Venn diagram starting from the intersection and working outward.
Problem. In a survey of 30 students, n(A) = 18 like art, n(B) = 15 like baking, and n(A ∩ B) = 8 like both. Find n(neither) and P(A only).
Step 1 — Place the intersection first.
n(A ∩ B) = 8 → write 8 in the overlap
Reason: the intersection is part of both n(A) and n(B), so we must subtract it before we can fill the "only" regions.
Step 2 — A only = n(A) − n(A ∩ B).
A only = 18 − 8 = 10
Reason: of the 18 people who like A, 8 also like B — so only 10 like A alone.
Step 3 — B only = n(B) − n(A ∩ B).
B only = 15 − 8 = 7
Step 4 — Neither = n(ξ) − (A only + A ∩ B + B only).
Neither = 30 − 10 − 8 − 7 = 5
Step 5 — Check all regions add to the total.
10 + 8 + 7 + 5 = 30 ✓
Step 6 — Calculate P(A only).
P(A only) = 10 / 30 = 1 / 3
Answer: n(neither) = 5 and P(A only) = 1/3.
2. We do — fill in the missing steps
Same shape as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. A class of 25 students surveyed: n(F) = 14 play football, n(N) = 10 play netball, n(F ∩ N) = 6 play both. Find n(neither) and P(F only).
Step 1 — Intersection in the overlap:
n(F ∩ N) = ______
Step 2 — F only = n(F) − n(F ∩ N):
F only = ______ − ______ = ______
Step 3 — N only = n(N) − n(F ∩ N):
N only = ______ − ______ = ______
Step 4 — Neither:
Neither = 25 − ______ − ______ − ______ = ______
Step 5 — Check:
______ + ______ + ______ + ______ = 25 ✓
Step 6 — P(F only):
P(F only) = ______ / 25 = ______
3. You do — independent practice
Show your working under each problem. Foundation problems are single-skill, standard apply the full method, and extension uses the addition rule.
Foundation — recall and basic regions
3.1 Write the formula for the addition rule for P(A ∪ B). 1 mark
3.2 If n(A) = 20 and n(A ∩ B) = 6, find "A only". 1 mark
3.3 Two events A and B are mutually exclusive. State the value of P(A ∩ B). 1 mark
3.4 n(ξ) = 40, A only = 12, n(A ∩ B) = 5, B only = 18. Find n(neither). 1 mark
Standard — fill the diagram and find a probability
3.5 In a group of 35 students, 20 like maths (M), 18 like science (S), and 9 like both. (a) Find M only, S only, and neither. (b) Calculate P(M ∩ S). 2 marks
3.6 n(ξ) = 60, n(A) = 25, n(B) = 30, n(A ∩ B) = 10. Find P(A ∪ B). 2 marks
Extension — addition rule and verification
3.7 Use the addition rule with P(A) = 0.45, P(B) = 0.30, P(A ∩ B) = 0.15 to find P(A ∪ B). Then state the value of P(neither). 2 marks
3.8 A class of 28 students: 16 own a dog (D), 12 own a cat (C), and 7 own neither pet. (a) How many own both? (b) Find P(owns a cat only). 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (football and netball)
Step 1: 6. Step 2: F only = 14 − 6 = 8. Step 3: N only = 10 − 6 = 4. Step 4: Neither = 25 − 8 − 6 − 4 = 7. Step 5: 8 + 6 + 4 + 7 = 25 ✓. Step 6: P(F only) = 8 / 25.
3.1 — Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
3.2 — A only
A only = n(A) − n(A ∩ B) = 20 − 6 = 14.
3.3 — Mutually exclusive
P(A ∩ B) = 0 — the events cannot both occur, so the circles do not overlap.
3.4 — Neither
Neither = 40 − 12 − 5 − 18 = 5.
3.5 — Maths and science
(a) M only = 20 − 9 = 11; S only = 18 − 9 = 9; Neither = 35 − 11 − 9 − 9 = 6. Check: 11 + 9 + 9 + 6 = 35 ✓.
(b) P(M ∩ S) = 9 / 35.
3.6 — P(A ∪ B)
Addition rule: P(A ∪ B) = 25/60 + 30/60 − 10/60 = 45 / 60 = 3/4.
3.7 — Addition rule with probabilities
P(A ∪ B) = 0.45 + 0.30 − 0.15 = 0.60. P(neither) = 1 − P(A ∪ B) = 1 − 0.60 = 0.40.
3.8 — Dogs and cats
(a) In at least one set = 28 − 7 = 21. By the addition rule, 21 = 16 + 12 − n(both), so n(both) = 28 − 21 = 7.
(b) C only = 12 − 7 = 5. P(cat only) = 5 / 28.