Probability
Probability questions cover conditional probability, independent events, total probability, Bayes' theorem, random variables with mean and variance, and the binomial distribution. Expect a 4–6 mark Bayes or distribution problem.
Formulas
- P(A|B) = P(A ∩ B)/P(B), P(B) > 0; P(A ∩ B) = P(A)P(B|A).
- A and B are independent if P(A ∩ B) = P(A)P(B).
- Total probability: P(E) = 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).
- Mean of a random variable: μ = Σ xᵢpᵢ; variance = Σ xᵢ²pᵢ − μ².
- Binomial: P(X = r) = ⁿCᵣ pʳ q^(n−r), mean np, variance npq.
Worked example
Box I has 2 white and 3 black balls; Box II has 4 white and 1 black. A box is chosen at random and a white ball is drawn. Find the probability it came from Box I. P(W) = ½(2/5) + ½(4/5) = 1/5 + 2/5 = 3/5. By Bayes, P(I|W) = (1/5)/(3/5) = 1/3.
Traps
- Confusing P(A|B) with P(B|A).
- Assuming independence without justification.
- Using with-replacement formulas when drawing is without replacement.
- Computing variance as Σx²p without subtracting μ².