Solutions — Events, Sets and Venn Diagrams

← Back to Questions

1. Q1 — Set notation for shaded regions

   (a) A only (excluding overlap): A ∩ B′   (b) Overlap only: A ∩ B   (c) Outside both circles: (A ∪ B)′   (d) Entire right circle including overlap: B

2. Q2 — Die sample space

   (a) A = {2,4,6}; B = {4,5,6}; A∩B = {4,6}; A∪B = {2,4,5,6}   (b) P(A) = 3/6 = 1/2; P(B) = 3/6 = 1/2; P(A∩B) = 2/6 = 1/3; P(A∪B) = 4/6 = 2/3   (c) P(A′) = 1 − 1/2 = 1/2; P(B′) = 1/2   (d) Not mutually exclusive: A∩B = {4,6} ≠ ∅. Both 4 and 6 are even AND greater than 3.

3. Q3 — Reading Venn diagram counts

   Total = 30. Sport (including both) = 12 + 5 = 17. Music (including both) = 5 + 9 = 14.   (a) P(Sport) = 17/30   (b) P(Music) = 14/30 = 7/15   (c) P(Sport ∩ Music) = 5/30 = 1/6   (d) P(Sport ∪ Music) = (12+5+9)/30 = 26/30 = 13/15   (e) P(Sport′) = 1 − 17/30 = 13/30

4. Q4 — Mutually exclusive events

   (a) Rolling a 6 is even; even numbers are not odd. Rolling 6 cannot be odd. Mutually exclusive.   (b) A red king exists (king of hearts, king of diamonds). Not mutually exclusive. Example: King of hearts is both red and a king.   (c) 15 is both a multiple of 3 and 5. Not mutually exclusive.   (d) January has 31 days, so being born in January means being born in a month with 31 days too. Not mutually exclusive.

5. Q5 — Coffee and tea Venn diagram

   Coffee only = 25 − 8 = 17. Tea only = 18 − 8 = 10. Both = 8. Neither = 40 − 17 − 8 − 10 = 5.   (a) Venn: C only=17, C∩T=8, T only=10, neither=5 (total=40).   (b) P(C ∩ T′) = 17/40   (c) P(C ∪ T) = (17+8+10)/40 = 35/40 = 7/8   (d) P(neither) = 5/40 = 1/8   (e) P(C′ ∩ T) = 10/40 = 1/4

6. Q6 — Working with given probabilities

   (a) P(A∪B) = P(A) + P(B) − P(A∩B) = 0.5 + 0.4 − 0.2 = 0.7   (b) P(A′) = 1 − 0.5 = 0.5   (c) P(A∩B′) = P(A) − P(A∩B) = 0.5 − 0.2 = 0.3   (d) P(A′∩B′) = 1 − P(A∪B) = 1 − 0.7 = 0.3   (e) Regions: A only=0.3, A∩B=0.2, B only=0.2, neither=0.3. Sum = 0.3+0.2+0.2+0.3 = 1.0 ✓

7. Q7 — Three-set Venn diagram: instruments

   n(G∪P∪D) = 30+24+20 − 12 − 8 − 6 + 4 = 52   (a) n(G∪P∪D) = 52   (b) None = 60 − 52 = 8   (c) G only = 30 − (12+8) + 4 − 4 ... Using inclusion: G only = 30 − 12 − 8 + 4 = 14. P(G only) = 14/60 = 7/30   (d) P only = 24 − 12 − 6 + 4 = 10. D only = 20 − 8 − 6 + 4 = 10. Exactly one = 14+10+10 = 34. P(exactly one) = 34/60 = 17/30

8. Q8 — Working backwards

   (a) P(A∪B) = P(A)+P(B)−P(A∩B) → 0.85 = 0.6+0.45−P(A∩B) → P(A∩B) = 0.20   (b) P(A′∩B) = P(B) − P(A∩B) = 0.45 − 0.20 = 0.25   (c) P(neither) = 1 − P(A∪B) = 1 − 0.85 = 0.15

9. Q9 — Health survey (three sets)

   n(H∪D∪O) = 90+70+60 − 30 − 25 − 20 + 10 = 155   (a) At least one condition: 155 patients   (b) None = 200 − 155 = 45 patients   (c) H only = 90 − 30 − 25 + 10 = 45. D only = 70 − 30 − 20 + 10 = 30. O only = 60 − 25 − 20 + 10 = 25. Exactly one = 45+30+25 = 100. P(exactly one) = 100/200 = 1/2   (d) P(H only) = 45/200 = 9/40

10. Q10 — Logical deduction

    (a) Mutually exclusive: P(A∪B) = P(A)+P(B) → 0.7 = 0.4 + P(B) → P(B) = 0.3   (b) P(A∪B) = p + 2p − p/3 = 3p − p/3 = 8p/3 = 0.8 → p = 0.8×3/8 = 0.3   (c) Exhaustive: P(A)+P(B)+P(C) = 1. Substituting: 2P(B)+P(B)+3P(B) = 6P(B) = 1 → P(B) = 1/6. P(A) = 2/6 = 1/3. P(C) = 3/6 = 1/2.