Combinatorics & counting
Ross Ch. 1
Permutations, combinations, multinomial coefficients, distinguishable/indistinct objects. Covers the poker, bridge, and committee problems.
Problems: 1–12, 19, 25, 33–35, 37–38
Basic probability & sample spaces
Khan Academy
Equally likely outcomes, axioms of probability, inclusion-exclusion principle. Geometric probability (uniform on interval/area).
Problems: 1–5, 13–14, 23–24, 29–30, 47–54
Conditional probability & independence
MIT OCW 6.041 L2–L3
P(A|B), total probability theorem, independence of events. The card, urn, and system reliability problems.
Problems: 55–69, 61–65, 90–96
Bayes' theorem (deep)
MIT OCW 6.041 L4
Bayes' rule with prior/posterior, sequential experiments, false positive problems, overlook probabilities. Multi-hypothesis Bayes.
Problems: 80–89, 91–100
Geometric probability
Ross Ch. 2
Continuous uniform distributions on geometric regions. Area-ratio method. Meeting-time and broken-stick problems.
Problems: 23–24, 29–30, 47–54
Discrete RVs, PMF, expectation & variance
Udemy current course
Your current Udemy course covers this well — lectures 69–84. Discrete probability, E[X], D²X, transformations Y=aX+b.
Problems: 1–8, 13, 15–20
Named discrete distributions
Udemy current course
Binomial, Geometric, Poisson, Bernoulli — also Hypergeometric (less covered). Lectures 79–88 handle this section well.
Problems: 2–3, 5, 8–12, 27–30
Continuous RVs, PDFs, CDFs
MIT OCW 6.041 L8–L10
Probability density functions, cumulative distribution, E[X] and D²X via integration. Problems 31–35, 41 require a calculus-based approach.
Problems: 31–35, 41–42
Uniform, Exponential & Normal distributions
Khan Academy + MIT L9
Uniform over [a,b], exponential with memoryless property, normal and z-scores, CLT approximations for Binomial/Poisson sums.
Problems: 36–40, 43–47
Joint PMFs & marginal distributions
Ross Ch. 6
Building full joint probability tables, deriving marginals by summing rows/columns. Urn and dice problems with 2 dependent variables simultaneously.
Problems: 1, 2, 3, 5, 6, 7, 8
Independence from joint distributions
MIT OCW 6.041 L12
Testing whether P(X=x, Y=y) = P(X=x)·P(Y=y) for all pairs. Problems 4, 5b, 6b, 7c, 8c, 9 all require this check explicitly.
Problems: 4, 5b, 6b, 7c, 8c, 9
Covariance & Pearson correlation
Ross Ch. 7 + MIT L16
Cov(X,Y) = E[XY] − E[X]E[Y], computing E[XY] from joint table, ρ = Cov/σₓσᵧ. Problems 5d, 6d, 7e, 8d all require full derivation from scratch.
Problems: 5d, 6d, 7e, 8d