Probability
What is tested
Probability covers conditional probability, independence, the multiplication theorem, the law of total probability, Bayes' theorem, and random variables with mean, variance and the binomial distribution.
Core rules
- P(A|B) = P(A∩B)/P(B) for P(B) > 0; multiplication form P(A∩B) = P(A|B)P(B).
- A and B are independent when P(A∩B) = P(A)P(B). Mutually exclusive events have P(A∩B) = 0 and cannot be independent unless one has probability 0.
- Total probability: P(A) = Σ P(Eᵢ)P(A|Eᵢ) over a partition of the sample space.
- Bayes: P(Eᵢ|A) = P(Eᵢ)P(A|Eᵢ) / Σ P(Eⱼ)P(A|Eⱼ).
- Binomial: P(X = r) = C(n,r)pʳq^(n−r), mean np, variance npq.
Worked example
Bag I holds 2 white and 3 black balls; Bag II holds 3 white and 2 black. A bag is chosen at random and one ball drawn; it is white. Then P(Bag I | white) = [(1/2)(2/5)] / [(1/2)(2/5) + (1/2)(3/5)] = (1/5)/(1/2) = 2/5. The denominator is the total probability of white; the numerator keeps only the Bag I branch.
Marks-losing traps
Computing P(B|A) when P(A|B) was asked; treating independent events as mutually exclusive; forgetting the C(n,r) coefficient in a binomial probability; using p rather than q in the failure exponent; ignoring the changed denominator when sampling without replacement.