Probability
Probability
This domain tests classical probability, conditional probability, independence, and Bayes-type updating. ASSEB questions often use drawing without replacement or selecting a source and then an outcome, so the sample space and the conditioning event must be written clearly.
Core rules
- For equally likely outcomes, P(E)=number of favourable outcomes / total outcomes.
- P(A∪B)=P(A)+P(B)−P(A∩B). Mutually exclusive events have P(A∩B)=0.
- Conditional probability: P(A|B)=P(A∩B)/P(B), provided P(B)>0. Multiplication theorem: P(A∩B)=P(A|B)P(B).
- A and B are independent if P(A∩B)=P(A)P(B), equivalently P(A|B)=P(A).
- Total probability: if B1,B2 partition the sample space, P(A)=ΣP(A|Bi)P(Bi). Bayes: P(Bi|A)=P(A|Bi)P(Bi)/P(A).
Worked example
Urn I contains 2 red and 3 blue balls; Urn II contains 4 red and 1 blue. An urn is chosen at random, then one ball is drawn. Find P(red) and P(Urn I | red). P(Urn I)=P(Urn II)=1/2. P(red|I)=2/5, P(red|II)=4/5. By total probability, P(red)=1/2·2/5+1/2·4/5=1/5+2/5=3/5. Then P(I|red)=[P(red|I)P(I)]/P(red)=(1/2·2/5)/(3/5)=(1/5)/(3/5)=1/3.
Traps
Do not confuse P(A|B) with P(B|A); Bayes reverses the conditioning. Without replacement, denominators change after the first draw. Mutually exclusive is not the same as independent: two non-impossible mutually exclusive events are dependent. When using a tree, multiply along branches and add across mutually exclusive branches.