Probability
Probability
The exam tests conditional probability, independence, the multiplication theorem, total probability and Bayes' theorem, plus random variables and the binomial distribution.
Core rules
- Conditional probability: P(A|B) = P(A∩B)/P(B), provided P(B) > 0.
- Multiplication theorem: P(A∩B) = P(A)P(B|A) = P(B)P(A|B).
- A and B are independent if P(A∩B) = P(A)P(B); mutually exclusive events with nonzero probability are not independent.
- Total probability: for a partition B₁, B₂, …, P(A) = ΣP(Bᵢ)P(A|Bᵢ). Bayes' theorem reverses the conditioning: P(Bᵢ|A) = P(Bᵢ)P(A|Bᵢ)/ΣP(Bⱼ)P(A|Bⱼ).
- Binomial distribution: P(X = r) = ⁿCᵣ pʳ qⁿ⁻ʳ, with mean np and variance npq.
Worked example
Bag I has 2 white and 3 black balls; Bag II has 3 white and 2 black balls. A bag is chosen at random and a white ball is drawn. Find the probability it came from Bag I.
P(I|W) = [P(I)P(W|I)]/[P(I)P(W|I) + P(II)P(W|II)] = [(1/2)(2/5)]/[(1/2)(2/5) + (1/2)(3/5)] = (1/5)/(1/2) = 2/5.
Traps
- Confusing P(A|B) with P(B|A).
- Treating mutually exclusive events as independent.
- Forgetting that without replacement changes the denominator on the next draw.
- Using the binomial formula when trials are not independent or n is not fixed.
- Mixing up mean np and variance npq.