Probability
What the exam tests
Probability questions use the classical definition, the addition and multiplication theorems, conditional probability and independence. Many are set as dice, cards, balls from urns, or business decision examples.
Core rules
- P(A) = number of favourable outcomes / total equally likely outcomes; 0 ≤ P(A) ≤ 1.
- P(A′) = 1 − P(A).
- Addition: P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for mutually exclusive events, P(A ∩ B) = 0.
- Multiplication: P(A ∩ B) = P(A)P(B|A); for independent events, P(A ∩ B) = P(A)P(B).
- Conditional: P(A|B) = P(A ∩ B)/P(B), P(B) > 0.
Worked example
Two fair dice are rolled. Find the probability that the sum is 7. Favourable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) - six outcomes out of 36. P(sum 7) = 6/36 = 1/6.
Conditional check: P(A) = 0.5, P(B) = 0.3, P(A ∩ B) = 0.15. P(A|B) = 0.15/0.3 = 0.5. Since P(A|B) = P(A) = 0.5, A and B are independent. Also P(A ∪ B) = 0.5 + 0.3 − 0.15 = 0.65.
Traps
- Assuming mutually exclusive events are independent; they are opposites in effect (if one occurs, the other cannot).
- Using P(A)P(B) for dependent events drawn without replacement.
- Double-counting the intersection in P(A ∪ B).
- Treating at least one as a single case instead of using 1 − P(none).
- Forgetting that the total changes when sampling without replacement.