Discrete distributions
Probabilities, mean and variance, the most likely value, expected payoffs, and the smallest sample size for a target.
Two or three events
Fill the 2×2 table from any three facts, test independence, and get the bounds when only P(A) and P(B) are known.
Total probability and Bayes’ rule
Hypotheses that partition the sample space, their prior probabilities, and the probability of the observed event under each.
Independent events with different probabilities
How many of the events occur: none, all, exactly j, at least j. Components in series need all to work; in parallel, at least one.
A probability table p(y)
E(Y), E(Y²), V(Y), σ, the expected value of any function g(Y), and Tchebysheff’s check. Leave one p(y) empty and it is filled with 1 − Σ.
Counting sample points
Factorials, permutations, combinations and partitions as exact whole numbers, however large, and an exact calculator for probabilities built from them.
Two-way table of counts
Joint, marginal and conditional probabilities, and whether the row and column classifications are independent.
Summarising data
Mean, variance with n − 1, standard deviation, the range/4 check, Tchebysheff’s theorem and the empirical rule (Chapter 1).
Results are computed in your browser with exact arithmetic where possible. Notation follows Wackerly, Mendenhall & Scheaffer, Mathematical Statistics with Applications, 7th ed. Report answers to the precision each question asks for.