Probability and Statistics
This domain covers conditional probability and Bayes' theorem, random variables, the binomial and Poisson distributions, and the statistics of dispersion: range, mean deviation, variance, standard deviation and coefficient of variation.
Core rules and concepts
- Conditional probability P(A|B) = P(A ∩ B)/P(B); the total probability theorem builds the denominator for Bayes' theorem.
- Bayes: P(Aᵢ|B) = P(Aᵢ)P(B|Aᵢ) / Σ P(Aⱼ)P(B|Aⱼ).
- Binomial: P(X = r) = nCr pʳ qⁿ⁻ʳ, mean np, variance npq, standard deviation √(npq).
- Poisson: P(X = r) = e^{−λ}λʳ/r!, mean = variance = λ, used when n is large and p small.
- Variance σ² = Σx²/n − x̄²; coefficient of variation = 100σ/x̄, a unit-free comparison of spread.
Worked example
Urn A holds 2 white and 3 red balls; urn B holds 4 white and 1 red. An urn is chosen at random and a white ball is drawn. Find the probability it came from A. P(A)P(W|A) = ½ × 2/5 = 1/5; P(B)P(W|B) = ½ × 4/5 = 2/5. Denominator = 1/5 + 2/5 = 3/5. P(A|W) = (1/5)/(3/5) = 1/3.
Traps that cost marks
- Confusing P(A|B) with P(B|A).
- Omitting the total-probability denominator in Bayes' theorem.
- Using npq as the mean, or forgetting the square root for standard deviation.
- Applying Poisson without checking that n is large and p small.