The School of Numbers & Logic · mathematics, pure and applied
Probability & Statistics
Reasoning honestly when certainty is unavailable — counting chances, weighing evidence, and knowing what a result does and does not show.
The rules of chance from counting to the normal curve, with the classic fallacies named and defused along the way.
Syllabus · 4 units · ~24 hours
Unit I — Chance, Counted
Sample spaces and events · Equally likely outcomes and the counting principle · Permutations and combinations · The birthday problem and other counting surprises
Unit II — The Rules of Probability
Addition and complement rules · Conditional probability · Independence, and what it does not mean · The gambler's fallacy examined
Unit III — Random Variables
Random variables and probability distributions · Expected value: the long-run average · Variance and standard deviation · Why casinos and insurers stay solvent
Unit IV — Named Distributions
The binomial distribution · The geometric distribution and waiting times · The normal curve and the empirical rule · The law of large numbers, observed
Study design, sampling, and inference — the full route from a question to a defensible conclusion, potholes marked.
Updating beliefs with evidence, by the rule that makes it precise — from medical tests to everyday judgment.
Syllabus · 4 units · ~16 hours
Unit I — The Rule Itself
Prior, likelihood, posterior · Bayes' theorem derived from a two-way table · Natural frequencies: the format that makes it easy
Unit II — Base Rates
The medical-test problem, worked in full · Base-rate neglect and why intuition fails · False positives in screening, security, and spam filters
Unit III — Updating in Sequence
Chaining evidence: today's posterior, tomorrow's prior · Strength of evidence and likelihood ratios · When evidence should barely move you
Unit IV — Bayes at Large
Choosing a prior honestly · Bayesian and frequentist answers compared on one problem · A/B comparisons and simple Bayesian estimation
A working defense against numbers used carelessly or cunningly — averages, charts, risks, and polls as they appear in the news.
Syllabus · 4 units · ~10 hours
Unit I — Averages & What They Hide
Mean versus median in skewed data · Simpson's paradox with a real example · Percentages of what, exactly
Unit II — Charts That Mislead
Truncated axes and inflated differences · Cherry-picked windows of time · Area and volume tricks in pictorial charts
Unit III — Risk & Causation
Relative versus absolute risk · Correlation, causation, and the missing third variable · Regression to the mean, or why streaky things settle down · Survivorship bias
Unit IV — Polls & Studies
Margins of error and what they cover · Who was asked, and who answered · Small samples and loud headlines · A checklist for reading any statistical claim
The theory behind the toolbox — estimators, likelihood, and the theorems that justify (and limit) statistical practice.
Syllabus · 5 units · ~44 hours
Unit I — Distribution Theory
Random variables, densities, and distribution functions · Joint, marginal, and conditional distributions · Transformations and moment generating functions · The gamma, chi-square, t, and F families
Unit II — Estimation
Estimators as random variables · Bias, variance, and mean squared error · Sufficiency and the Rao–Blackwell theorem · The Cramér–Rao lower bound
Unit III — Likelihood
The likelihood function · Maximum likelihood estimation and its asymptotics · Fisher information
Unit IV — Testing
The Neyman–Pearson lemma · Likelihood ratio tests · Power and sample size, formally · Multiple testing and its corrections
Unit V — Regression Foundations
The linear model in matrix form · Least squares and the Gauss–Markov theorem · Inference for coefficients · What the model assumes and how assumptions fail