Combinatorics and Probability: From Counting to Inference

Learn the fundamental Rule of Product for counting multi-stage processes and verify calculations with Python.

Learn when to add vs multiply in counting, understand permutations and factorial notation.

Master unordered selection using the binomial coefficient, Pascal's triangle, and combinatorial identities.

Master multisets, stars and bars, and counting with indistinguishable objects or repeated selections.

Master counting with overlapping sets using inclusion-exclusion, and apply it to derangements and constraint problems.

Master the pigeonhole principle for existence proofs, worst-case guarantees, and surprising mathematical results.

Master problem-solving strategies including symmetry, bijections, double counting, and combinatorial proofs.

Learn to encode sequences as power series and solve counting problems algebraically.

Apply generating functions to integer partitions, coin change problems, and convolutions.

Master EGFs for labeled structures, derangements, and exponential series.

Solve recurrence relations using the characteristic equation method and verify with Python.

Master bijective proofs, double counting, and understand Catalan numbers.

Understand the foundations of probability: sample spaces, events, and event algebra.

Master Kolmogorov's axioms and derive fundamental properties including complement rule and addition rule.

Apply counting techniques to calculate probabilities on finite uniform spaces.

Master conditional probability, multiplication rule, and reduced sample spaces.

Master the law of total probability and Bayes' theorem for probabilistic inference.

Master independence of events, distinguish pairwise from mutual independence, and test independence in practice.

Master Monty Hall, Simpson's paradox, and other counterintuitive conditional probability problems.

Master discrete random variables, PMFs, CDFs, and their properties with Python implementations.

Learn common discrete distributions.

Learn continuous random variables.

Learn the normal distribution.

Learn expectation and linearity of expectation.

Learn variance, standard deviation, and covariance.

Learn moment generating functions.

Learn joint, marginal, and conditional distributions.

Learn laws of total expectation and variance.

Master balls in bins and occupancy problems with applications in probability and combinatorics.

Master birthday paradox and collision problems with applications in probability and combinatorics.

Master coupon collector and random graphs with applications in probability and combinatorics.

Master introduction to the probabilistic method with applications in probability and combinatorics.

Master linearity of expectation in combinatorics with applications in probability and combinatorics.

Master alteration method and lovász local lemma with applications in probability and combinatorics.

Master law of large numbers with applications in probability and combinatorics.

Master central limit theorem with applications in probability and combinatorics.

Master markov and chebyshev inequalities with applications in probability and combinatorics.

Master chernoff and hoeffding bounds with applications in probability and combinatorics.

Master randomized algorithms fundamentals with applications in probability and combinatorics.

Master hashing and load balancing with applications in probability and combinatorics.

Master bloom filters and probabilistic data structures with applications in probability and combinatorics.

Master probability in financial modeling with applications in probability and combinatorics.

Master portfolio theory and risk metrics with applications in probability and combinatorics.

Master bayesian inference and naive bayes classifier with applications in probability and combinatorics.

Master markov chains and monte carlo methods with applications in probability and combinatorics.

Master probabilistic paradoxes and counterintuitive results with applications in probability and combinatorics.

Master fallacies in statistical reasoning with applications in probability and combinatorics.

Master problem solving strategies in probability with applications in probability and combinatorics.

Master capstone project design with applications in probability and combinatorics.

Master capstone project implementation with applications in probability and combinatorics.