Probability of Combined Events
8.3 Probability of Combined Events – Cheat Sheet
1. Notation and meanings
- P(A ∩ B) → Probability of A and B occurring (intersection)
- P(A ∪ B) → Probability of A or B occurring (union)
- P(A′) → Probability of not A
- With replacement → Probability stays the same for each trial
- Without replacement → Probability changes after each trial
2. Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
3. Multiplication rule
- Independent events: P(A ∩ B) = P(A) × P(B)
- Dependent events: P(A ∩ B) = P(A) × P(B|A)
4. Tree diagrams
- Write outcomes at the end of each branch
- Write probabilities along each branch
- Multiply probabilities along a path for P(A ∩ B)
- Add probabilities of different paths for P(A ∪ B)
5. Venn diagrams
| Region | Probability representation |
|---|---|
| A only | P(A) − P(A ∩ B) |
| B only | P(B) − P(A ∩ B) |
| Both A and B | P(A ∩ B) |
| Neither A nor B | 1 − P(A ∪ B) |
6. Sample space diagrams
- List all possible outcomes
- Count favourable outcomes / total outcomes
Example: Toss 2 coins
S = {HH, HT, TH, TT}
P(2 heads) = 1/4
P(at least 1 head) = 3/4
7. Examples
| Event | P(Event) |
|---|---|
| P(A) | 0.5 |
| P(B) | 0.3 |
| P(A ∩ B) | 0.1 |
P(A ∪ B) = 0.5 + 0.3 − 0.1 = 0.7