Foundation of quantitative reasoning: counting, conditional probability, distributions, and stochastic models.
Permutations, combinations, and the multiplication principle
P(A|B) and the chain rule of probability
Updating beliefs with new evidence
Random variables, linearity of expectation, and weighted averages
Measuring spread, key inequalities, and the birthday problem
Law of iterated expectations, law of total variance, and multi-stage problems
Dice probability, stopping games, and expected values with dice
Biased coins, streaks, and head/tail problems
Poker hands, drawing without replacement, and card game probabilities
Binomial, geometric, Poisson, and dice problems
Uniform random variables, exponential waiting times, and competing exponentials
Gaussian properties, z-values, gamma waiting times, and log-normal prices
Poisson arrivals, merging, and thinning
MGFs, deriving moments, and sums of independent variables
Expected utility, risk aversion, and first/second-order stochastic dominance
Drawing without replacement
Chernoff bounds, Hoeffding inequality, and exponential tail estimates
Min, max, k-th order statistic, Beta connection, exponential spacings, and range
State transitions, steady states, and random walks on graphs
Hitting times, return times, and walks on cubes, cycles, and lattices
Optimal stopping, fair games, bet sizing, and tournament probability
When to commit, the 1/e rule, and value-based stopping thresholds