Probability

Why counting equally likely cases gives real numbers and then breaks, what a long record of coin tosses does and does not settle, and the short list of rules every reading of probability agrees on.

The sample space of everything that can happen, events as subsets of it, the three axioms that replace a definition, and the rules that follow from them by set algebra alone.

The product rule, permutations and combinations, tested on poker hands and on the birthday problem, where a correct count contradicts almost everyone's intuition.

What learning something does to a sample space, the chain rule that follows, a definition of independence that survives contact with examples, and the law of total probability.

Reversing a conditional probability, why a positive result on an accurate test usually means nothing, the odds form that makes updating a multiplication, and what the theorem cannot supply.

Attaching a number to every outcome, the distribution that results, expectation as a weighted average, and the linearity that solves problems no direct sum can reach.

Measuring how far a random variable strays from its mean, why the measure is built from squares, why variances add for independent variables, and the guarantee Chebyshev extracts from a variance alone.

The first named law: counting successes in a fixed number of independent trials, its mass function from the counting lesson, its mean and variance from indicators, and the waiting time that comes with it.

The limit of the binomial when trials are many and success is unlikely, a distribution with one parameter and equal mean and variance, tested against horse kicks, bomb maps and alpha particles.

Replacing the sum with an area and the mass function with a density, why every single value has probability zero without being impossible, and the two continuous laws that follow directly from earlier lessons.

The one distribution that keeps appearing: its formula and its two parameters, standardisation to a single table, the 68, 95 and 99.7 figures computed rather than recited, and what it is a bad model of.

Proving that averages settle down, distinguishing what that does and does not promise, and the theorem that says the leftover error is normally distributed whatever the ingredients were.

Six standard failures of probabilistic reasoning, each traced to the rule it violates: base rate neglect, the prosecutor's fallacy, regression to the mean, multiple comparisons, survivorship and coincidence.