Probability (P) Exam
Probability foundations
The Probability exam rewards precise use of axioms and conditioning. Know set operations, independence, mutually exclusive events, conditional probability, Bayes theorem, and the law of total probability. For random variables, work with probability mass functions, density functions, cumulative distribution functions, support, expectation, variance, and moment generating functions. For joint distributions, derive marginal and conditional distributions and check independence by factoring the joint density or mass function.
Common traps
Do not confuse independence with mutual exclusivity: mutually exclusive events with positive probability are dependent. When using a density, integrate over the correct support and remember that a single point has probability zero. In conditional variance, use Var(X) = E[Var(X|Y)] + Var(E[X|Y]). For the central limit theorem, state whether you are standardizing a sum or a sample mean and use the correct standard error. Check whether sampling is with or without replacement.
Worked rule
A disease has prevalence 1%. A test is positive for 90% of diseased people and 5% of healthy people. Then P(disease | positive) = (0.01)(0.90) / [(0.01)(0.90) + (0.99)(0.05)] = 0.009/(0.0585) = 15.4%. The base rate matters. A common error is answering 90%, which reverses the conditional probability. Draw a tree or two-way table to keep the conditioning direction clear.