Probability and Statistics
Probability and statistics practice develops conditional reasoning, standard distributions, and correct interpretation of inference - the last of which is where most marks are lost.
Probability toolkit
- P(A∪B) = P(A) + P(B) − P(A∩B); independence means P(A∩B) = P(A)P(B).
- Conditional probability P(A|B) = P(A∩B)/P(B); Bayes reverses the conditioning: P(A|B) = P(B|A)P(A)/P(B).
- Law of total probability: sum over a partition of the sample space.
- E[aX + b] = aE[X] + b; Var(aX + b) = a²Var(X); Var(X) = E[X²] − (E[X])².
Distributions the paper uses
- Binomial(n, p): mean np, variance np(1 − p).
- Poisson(λ): mean and variance both λ, used for counts.
- Normal: standardise with z = (x − μ)/σ before using tables.
- Sampling: X̄ has mean μ and standard error σ/√n; for independent identically distributed observations with finite variance, the CLT supports a normal approximation to the standardized sample mean as n grows; the required sample size depends on population shape.
Inference
- A confidence interval describes the procedure, not a probability attached to a fixed parameter.
- A p-value is P(data at least as extreme | H₀), not P(H₀ true).
- Test statistic for a mean with known σ: z = (x̄ − μ₀)/(σ/√n).
- Unbiasedness, efficiency and consistency are estimator properties; the sample variance divides by n − 1 to be unbiased.
Worked example
A disease affects 1% of a population. A test has sensitivity 0.95 and specificity 0.90. Given a positive test, find P(disease). P(+) = 0.95(0.01) + 0.10(0.99) = 0.0095 + 0.099 = 0.1085. P(D|+) = 0.0095 / 0.1085 ≈ 0.0876, about 8.8%. The low prior dominates - the base-rate effect.
Traps that cost marks
- Confusing P(A|B) with P(B|A).
- Using σ instead of σ/√n, or dropping the √n entirely.
- Treating 'fail to reject H₀' as proof that H₀ is true.
- Adding variances of dependent random variables.
- For discrete data, using the wrong tail or omitting a continuity correction.