Venn Diagrams & Set Notation
Venn Diagrams & Set Notation
Venn diagrams use overlapping circles to show how groups (sets) relate. They are a powerful tool for probability, especially when events overlap.
Set Notation
| Symbol | Meaning | Example |
|---|---|---|
| ξ (xi) | Universal set (everything) | All students in a class |
| A | Set A | Students who play football |
| A ∩ B | A AND B (intersection) | Students in both A and B |
| A ∪ B | A OR B (union) | Students in A or B or both |
| A' | NOT A (complement) | Students not in A |
| n(A) | Number of elements in A | n(A) = 15 means 15 members |
| ∅ | Empty set | No members |
Drawing Venn Diagrams from Information
Worked Example: In a class of 30 students, 18 study French (F), 14 study German (G), and 8 study both.
Method: Always fill in the intersection first.
1. Both: F ∩ G = 8
2. French only: 18 − 8 = 10
3. German only: 14 − 8 = 6
4. Neither: 30 − 10 − 8 − 6 = 6
Check: 10 + 8 + 6 + 6 = 30 ✓
Reading Probabilities from Venn Diagrams
Using the example above (30 students total):
- P(F) = 18/30 = 3/5
- P(F ∩ G) = 8/30 = 4/15 (both)
- P(F ∪ G) = (10 + 8 + 6)/30 = 24/30 = 4/5 (French or German or both)
- P(F') = 12/30 = 2/5 (not French)
- P(F ∩ G') = 10/30 = 1/3 (French only, not German)
- P(F' ∩ G') = 6/30 = 1/5 (neither)
Three-Set Venn Diagrams (Higher)
With three sets, there are 8 regions. Fill from the innermost region (all three) outward.
Worked Example: 50 people were asked about pets. 25 have dogs (D), 20 have cats (C), 15 have fish (F). 8 have dogs and cats, 5 have dogs and fish, 4 have cats and fish, and 2 have all three.
Fill in order:
1. All three: D ∩ C ∩ F = 2
2. Dogs and cats only: 8 − 2 = 6
3. Dogs and fish only: 5 − 2 = 3
4. Cats and fish only: 4 − 2 = 2
5. Dogs only: 25 − 6 − 2 − 3 = 14
6. Cats only: 20 − 6 − 2 − 2 = 10
7. Fish only: 15 − 3 − 2 − 2 = 8
8. None: 50 − 14 − 6 − 3 − 2 − 10 − 2 − 8 = 5
Conditional Probability from Venn Diagrams (Higher)
P(A | B) = P(A ∩ B) / P(B) = n(A ∩ B) / n(B)
Worked Example: Using the French/German example, find P(French | German).
- n(F ∩ G) = 8
- n(G) = 14
- P(F | G) = 8/14 = 4/7
This means: of the students who study German, 4/7 also study French.
Shading Venn Diagrams
Exam questions may ask you to shade regions described in set notation:
- A ∩ B: shade ONLY the overlap
- A ∪ B: shade everything in A or B (including overlap)
- A' ∩ B: shade the part of B that is NOT in A
- (A ∪ B)': shade everything OUTSIDE both circles
Exam Tips
- ALWAYS start with the intersection when filling in a Venn diagram — working from the outside in gives wrong numbers
- For three sets, start with the triple intersection, then double intersections, then single regions
- P(A ∪ B) is NOT P(A) + P(B) unless the events are mutually exclusive — use the Venn diagram to avoid double counting
- For conditional probability from a Venn diagram, restrict your attention to one circle only — that circle's total becomes your denominator