Probability and Statistics
What to master
Know the standard distributions and when each applies: binomial (fixed n, independent trials, constant p), Poisson (counts per unit time or space), normal (with z-scores and the empirical 68/95/99.7 rule), exponential (time between Poisson events), and the hypergeometric idea of sampling without replacement. Be fluent in combinations and permutations, conditional probability, independence, Bayes' theorem, expected value and variance, standard error, confidence intervals, hypothesis testing (Type I vs Type II error, p-value logic), and simple linear regression (slope, correlation coefficient, coefficient of determination).
Traps
- P(A|B) is not P(B|A); Bayes questions hinge on reversing them correctly.
- Sample standard deviation divides by (n - 1); population divides by n. The exam will offer both.
- "At least one" is almost always easiest via the complement: 1 - P(none).
- Without-replacement sampling changes p from draw to draw and invalidates the simple binomial formula.
- Confidence level is not the probability that a computed interval contains the parameter in repeated sampling -- wording matters.
Worked example
A process yields 10% defective parts; three parts are drawn independently. P(at least one defective) = 1 - P(all good) = 1 - 0.9^3 = 1 - 0.729 = 0.271.
Rule of thumb: write down what is fixed (n, p) and whether sampling is with replacement before touching a formula; and convert every normal problem to a z-score with sigma of the mean, sigma/sqrt(n), when working with sample averages.