Probability and Statistics (Subtest II)
Counting and Probability Rules
Probability and Statistics on Subtest II tests counting, probability models, and inference. Distinguish permutations (order matters, nPr = n!/(n−r)!) from combinations (order ignored, nCr = n!/(r!(n−r)!)). For events A and B: P(A∪B) = P(A) + P(B) − P(A∩B); if independent, P(A∩B) = P(A)P(B); conditional probability is P(A|B) = P(A∩B)/P(B). Bayes' theorem reverses a conditional: P(A|B) = P(B|A)P(A)/P(B).
Random Variables and Distributions
A discrete random variable has expected value E(X) = Σx·P(x) and variance E(X²) − [E(X)]². Know the binomial distribution (fixed n, independent trials, constant p): P(X = k) = C(n,k)p^k(1−p)^{n−k}, with mean np; the geometric distribution for the first success, with mean 1/p; and the normal distribution with the empirical 68-95-99.7 rule and z-scores z = (x − μ)/σ.
Inference and Data
Sampling methods (simple random, stratified, cluster, systematic) determine whether conclusions generalize. The sampling distribution of the mean has standard error σ/√n, and a confidence interval is estimate ± critical value × standard error. In a hypothesis test the p-value is the probability of data at least as extreme as observed, assuming the null hypothesis is true; it is not the probability that the null is true.
Worked Example
A disease affects 1% of a population. A test is 95% sensitive and 90% specific. Given a positive test, what is P(disease)? P(disease and positive) = 0.01 × 0.95 = 0.0095. P(no disease and positive) = 0.99 × 0.10 = 0.099. So P(positive) = 0.1085, and P(disease | positive) = 0.0095/0.1085 ≈ 0.0876, about 8.8%. Despite the high sensitivity, most positives are false, because the base rate is low.
Traps That Cost Marks
- Confusing P(A|B) with P(B|A).
- Adding probabilities for non-mutually-exclusive events without subtracting the overlap.
- Treating without-replacement draws as independent.
- Interpreting a p-value as the probability that H₀ is true.
- Concluding causation from a strong correlation, or extrapolating beyond the data range.