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Probability tutor

Counting

Combinatorial counting techniques that underlie the naive definition of probability: the multiplication rule, permutations and combinations, binomial coefficients, and the more specialized tools (sampling table, complement, stars and bars, inclusion-exclusion) needed for harder counting problems. Topics: Fundamental Counting Rules, Advanced Counting Techniques.

Probability Axioms & Rules

The formal foundations of probability: sample spaces and events, the naive definition, the Kolmogorov axioms and the properties they imply, odds, simulation as a way to estimate probabilities, and classic application problems (birthday, matching). Topics: Sample Spaces, Axioms, and Properties, Simulation and Classic Problems.

Conditional Probability & Independence

Conditioning on information: the definition and multiplication rule, the law of total probability and Bayes' rule for updating beliefs, the base-rate fallacy, independence (including pairwise vs. Topics: Conditional Probability and Bayes' Rule, Independence, Case Studies: Gambler's Ruin, Monty Hall, Simpson's Paradox.

Discrete Random Variables

Discrete random variables and their distributions: the RV concept, PMF and CDF, expectation and variance (including linearity, indicator RVs, and LOTUS), and the named discrete distributions (Bernoulli, binomial, hypergeometric, geometric, negative binomial, Poisson). Topics: Random Variables, PMF/CDF, Expectation, Variance, Named Discrete Distributions.

Continuous Random Variables

This unit extends random variables from discrete to continuous, replacing sums with integrals: densities and CDFs, expectation, variance, and LOTUS by integration, the uniform distribution and the change-of-variables technique it motivates, the memoryless exponential, and the normal distribution with its standardization trick, closing with a light look at the gamma distribution. Topics: Density Functions, Moments, and LOTUS, Uniform Distribution and Transformations, Exponential Distribution and Memorylessness, Normal Distribution and a Light Look at Gamma.

Joint Distributions

This unit handles more than one random variable at once: joint, marginal, and conditional distributions and the independence test built from them; covariance and correlation as measures of joint linear behavior feeding into the variance of a sum; conditional expectation and the tower and total-variance identities built on it; and a light look at the multinomial and bivariate normal as named joint distributions. Topics: Joint, Marginal, and Conditional Distributions, Sums, Covariance, and Variance of Sums, Conditional Expectation and the Tower Property.

Limit Theorems & Preview

This closing unit steps back from individual distributions to what happens in the limit: Markov's and Chebyshev's inequalities bound tails from just a mean or variance, the law of large numbers and Central Limit Theorem describe how sums and means behave as the sample size grows, moment generating functions package all of a distribution's moments into one tool, and a compact preview of the Poisson process and Markov chains previews where probability goes next. Topics: Inequalities and Limit Theorems, Moment Generating Functions, Stochastic Processes Preview.