Probability Theory and Distributions
This domain tests the rules that turn uncertain information into numbers: the probability axioms, conditional probability and Bayes, random variables with their expectations, and the standard distributions that later feed inference.
Probability rules
- Three axioms: probability is non-negative; the sample space has probability 1; for disjoint events the probability of the union is the sum of the probabilities.
- General addition rule: P(A or B) = P(A) + P(B) minus P(A and B). Forgetting to subtract the overlap double-counts.
- Multiplication rule: P(A and B) = P(A) times P(B given A). Independence means P(B given A) equals P(B), so the product simplifies.
- Mutually exclusive and independent are opposite in spirit: disjoint events with positive probability are never independent.
Conditional probability and Bayes
- P(A given B) = P(A and B) divided by P(B).
- Bayes theorem combines the test result with the base rate: P(D given +) = P(+ given D)P(D) divided by the total probability of a positive result. When a condition is rare, the base rate dominates.
Random variables
- E(aX + b) = aE(X) + b, while Var(aX + b) = a-squared times Var(X).
- Variances add for independent variables: Var(X minus Y) = Var(X) + Var(Y).
- Binomial(n, p) has mean np and variance np(1 minus p), for fixed n independent trials with constant p.
- Poisson with parameter lambda has mean equal to variance equal to lambda, for counts of rare independent events in a fixed interval.
- Normal probabilities come from standardising, z = (x minus mu) divided by sigma; add or subtract 0.5 as a continuity correction when approximating a discrete count.
Worked example
A condition has prevalence 2%, test sensitivity 95% and specificity 90%. The probability of a positive result is 0.95 times 0.02 plus 0.10 times 0.98 = 0.019 + 0.098 = 0.117. Then P(D given +) = 0.019/0.117 = 0.162. Despite a 95% sensitive test, only about 16% of positives are true cases, because false positives from the large healthy group swamp the true positives.
Traps
- Reporting the sensitivity as if it were the predictive value of a positive test.
- Using the variance where the standard deviation belongs in the z transformation.
- Writing Var(X minus Y) as Var(X) minus Var(Y).
- Applying Poisson when events cluster, which makes the variance larger than the mean.