Probability theory

The mathematics of chance: a sample space of possible outcomes, a measure assigning weight to each, and the machinery for reasoning about subsets of it. The corpus holds two accounts. mcmullen-probability-theory is a 98-page companion to Feller’s 1950 textbook rather than a standalone treatment; stirzaker-elementary-probability (added 2026-08-06) is a 538-page Cambridge course that assumes nothing but set theory and carries its own exercises, problems and solutions. They agree about the mathematics and differ about how much of it a reader is expected to supply.

The discrete foundation

The starting definitions are set-theoretic and deliberately unglamorous. A sample space S; an event is a subset of it; logical combinations of events are the operators of set theory (A′ = S − A, A ∩ B, A ∪ B). A probability measure on a discrete space is a function p : S → [0,1] summing to 1, and P(A) = Σ_{s∈A} p(s). Where symmetry or ignorance applies, p(s) = 1/|S| and probability reduces to counting.

McMullen makes a point of deriving even the obvious from this: P(A′) = 1 − P(A) “is not based on intuition (although it coincides with it), but is derived from the definitions.” Which is why the next thing after the definitions is combinatorics — with equal weights, every probability question becomes a counting question, and the binomial coefficient C(n,k) does most of the work.

Where the subject gets its teeth

The material after counting is what makes it a theory rather than a technique: random walks, conditional probability, the binomial and Poisson distributions, normal approximation, unlimited sequences of Bernoulli trials, expectation, the law of large numbers, generating functions, and ruin problems. Then the continuous side — the exponential and uniform densities, and randomization.

Generating functions are the representative trick: to get the mean number of coin flips until the first head, write f(x) = Σ xⁿ/2ⁿ and differentiate, giving E = f′(1) = 2. An infinite sum becomes a derivative.

Specifying the sample space is the whole difficulty

The corpus’s clearest teaching point, and the one worth carrying into applied work. The mailman paradox in mcmullen-probability-theory: n letters delivered “at random” gives a 1/e ≈ 37% chance nobody gets the right one, which is flatly wrong for n = 2 where the answer is 50%. Nothing is wrong with the arithmetic. “At random” named two different spaces — all functions from letters to boxes, versus all bijections — and the formula belonged to one of them.

The same lesson underneath the birthday problem, Benford’s law and the significance question the notes open with: among 1,000 coins, one showing ten heads in a row is not evidence of a two-headed coin. What counts as the space of possibilities decides the answer before any calculation starts.

Where the subject continues: stochastic processes

stirzaker-elementary-probability carries the material past the point McMullen stops, and the extension has a name — a stochastic process is a family of random variables indexed by time, so the question moves from “what is the distribution” to “what does the sequence do.”

Martingales — a process whose expected next value, given everything so far, is its current value: the formalisation of a fair game. New to Stirzaker’s second edition, together with optional stopping, the result about when you may stop a fair game and still expect to break even.
Markov chains (ch. 9) — the process whose future depends on the present and not the path to it. Transition probabilities, first-passage times, stationary distributions, and what happens in the long run.
The Poisson process and, in continuous time, the Wiener process and diffusions — Brownian motion as the limit object, which is where the subject stops being about dice.

This is the layer under half the hub’s applied stochastics, and it arrived here in the same book that explains why: the second edition added Brownian motion because it underpins the Black–Scholes option formula.

Where this is used across the hub

Probability is the most-borrowed body of theory in the hub, and this page is the subject’s home rather than any of its applications.

../machine-learning-wiki — bayesian-inference is this measure applied to model parameters; monte-carlo-methods samples from distributions this theory defines.
../quant-trading-wiki — quant-bible-mit-sloan teaches conditional probability, Bayes, expectation and variance as interview material. Ruin problems, covered by McMullen, are the sharpest genuine overlap: they are the mathematics of going broke. Since 2026-08-06 the overlap is wider: stirzaker-elementary-probability‘s martingales, optional stopping and Wiener process are the machinery under option pricing, which that spoke uses and does not teach.
../optimization-algorithms-wiki — the stochastic search in every metaheuristic there rests on random walks.

mcmullen-probability-theory · stirzaker-elementary-probability · mathematics · mathematical-statistics · fourier-analysis · synthesis