Statistics and Probability
This domain tests reasoning under uncertainty: translating a story into a probability model, choosing an estimator or test, and reading the result without overclaiming.
Probability foundations
- Kolmogorov axioms; P(A union B) = P(A) + P(B) - P(A intersect B); independence means P(A intersect B) = P(A)P(B).
- Conditional probability P(A|B) = P(A intersect B)/P(B), giving Bayes: P(A|B) = P(B|A)P(A)/P(B).
- Expectation and variance: Var(aX) = a squared times Var(X); Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y).
- Distributions: Bernoulli, binomial, Poisson (counts, mean equals variance), normal, exponential (memoryless).
Inference
- The central limit theorem gives the sample mean an approximately normal sampling distribution with standard error sigma/root-n: the spread of the estimate, not of the raw data.
- A 95% confidence interval is a procedure: across repeated samples, 95% of such intervals cover the fixed parameter.
- A p-value is the probability of data at least as extreme as observed, given H0 is true. It is not the probability that H0 is true.
- Type I error alpha, Type II error beta, power 1 minus beta; power grows with sample size, effect size, and lower variance. Testing many hypotheses inflates false positives, so apply Bonferroni or Benjamini-Hochberg.
Worked example
Prevalence 1%, sensitivity 99%, specificity 95%. P(+) = 0.99 x 0.01 + 0.05 x 0.99 = 0.0099 + 0.0495 = 0.0594. Then P(D|+) = 0.0099 / 0.0594 = 0.167. Even a good test mostly returns false positives at a low base rate, so the posterior must be reported alongside sensitivity and specificity.
Traps
Reading a p-value as evidence about H0's probability; treating a confidence interval as a probability statement about a fixed parameter; confusing standard deviation with standard error; and comparing many groups without multiplicity correction.