The Mathematics of ChanceCHANCE / MATHEMATICS

The Mathematics of Chance, book cover
AN ORIGINAL BOOK ON PROBABILITY

The Mathematics
of Chance

The history, ideas and mathematical
structures of probability

The Mathematics of Chance

By Haobai Wang

From an interrupted game to random paths: follow the problems that made new mathematical ideas necessary, with sustained arguments, counterexamples and proofs.

31 long-form chapters · Work in progress

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完整英文原稿 / Full English manuscript
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Complete English translations of all 31 chapters and their proof materials. A working translation for author review, with the Chinese manuscripts preserved independently.

PRIMARY AUTHOR

Haobai Wang

Haobai Wang is the primary author of The Mathematics of Chance, directing its questions, narrative, mathematical structure and editorial adoption. The book follows historical problems to explain why probability concepts became necessary, moving toward modern probability and random geometry. It seeks breadth, depth and correct mathematical statements while preserving sustained arguments, counterexamples and narrative continuity.

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Each chapter opens its complete English text. Switch between section and full-chapter reading at any time.

Volume I · Can Chance Be Calculated?

Chapters 1–4

01An Unfinished GameFairness, symmetry, independence and additivity, beginning with an interrupted game.02From Chance to Knowledge: Learning an Unknown ProbabilityFrequencies, Bernoulli, Bayes and the problem of inference.03The Birth of the Bell Curvede Moivre, Laplace, Gauss and the emergence of Gaussian limits.04Randomness Enters Time: From Random Walks to Brownian MotionSquare-root scaling, joint distributions, covariance and the question of a random path.

Volume II · Probability in an Infinite World

Chapters 5–14

05The Price of Infinity: Why Probability Needs Measure TheoryFrom the difficulties of infinite sets to sigma-algebras, outer measures and probability.06The Rebirth of the Mean: From Weighted Sums to Lebesgue IntegrationIntegration, expectation and the conditions for exchanging a limit with an integral.07What Does Convergence Mean? Four Limits in a Random WorldRare recurring events, moving windows, spikes and uniform integrability.08How Can We Construct an Infinite Random Experiment?Consistent finite-dimensional rules, measure extension and a common space of records.09After Seeing Part of the World: Conditional Expectation and InformationFair prices, observable information, orthogonal prediction and Bayesian updating.10How Can One Probability Be Decomposed in Terms of Another? Densities, Conditional Laws and KernelsDensities, singular mass, regular conditional laws, disintegration and sequential experiments.11How Can a Distribution Be Recorded by a Function? Characteristic Functions, Independent Sums and Gaussian LimitsBounded frequency readings, inversion, Lévy continuity and a complete finite-variance central limit proof.12When Can Infinitely Many Additions Still Have an Answer?First crossings, independent series, three-series criteria, integrable strong laws and tail information.13Why Do Rare Events Produce New Stable Laws?Rare counts, Le Cam coupling, independent marks, Laplace transforms and the lattice local central limit theorem.14Not Every Limit Is Gaussian—Triangular Arrays, Stable Laws and Infinite DivisibilityChanging independent rows, Lindeberg–Feller, multivariate limits, stable attraction and infinite divisibility.

Volume III · Fairness, Memory and Regeneration in Time

Chapters 15–22

15What Becomes of a Game That Remains Fair Forever?Fair capital, maximal and crossing estimates, closed martingales, Lp convergence and decreasing information.16What If the Player Chooses When to Stop?Stopping rules, bounded sampling, UI settlement and gambling counterexamples.17Does the Present Suffice to Determine the Future?Markov kernels, sequential construction, strong Markov and stationary structure.18Why Do Some Random Walks Return While Others Escape?Returns, Green functions and the dimensional fate of random walks.19Waiting for a New BeginningRenewal equations, key renewal, length bias and stationary waiting.20How Does One Ancestor Become a Random Family?Extinction, population martingales, spines and the x log x condition.21Waiting without Memory—The Poisson ProcessExponential waits, counting paths, independent windows, compensated martingales and split or merged streams.22How Can a Random Process Jump?—Continuous-Time Markov ChainsState-dependent clocks, minimal jump chains, explosion, Kolmogorov equations, invariant laws and timed branching.

Volume IV · From Discrete Records to Continuous Paths

Chapters 23–29

23How Can Gaussian Snapshots Become Continuous Paths?—Continuous Modifications, Gaussian Processes and Wiener MeasureGaussian covariance, countable-grid continuity, Wiener measure, bridges, Ornstein–Uhlenbeck motion and isonormal geometry.24How Can an Entire Random Path Converge?—Portmanteau, Prokhorov and Function-Space TopologiesFrom missed spikes to Polish weak limits, coupling, continuous-path tightness and jump-time alignment.25When Does a Random Walk Become Brownian Motion?—Donsker’s Invariance PrincipleFinite-variance path limits through fixed truncation, maximal estimates, empirical bridges and inverse renewal clocks.26Why Is a Continuous Path So Rough?From first-hit reflection to nowhere differentiability, exact oscillation, functional LIL and arcsine occupation laws.27What If Random Motion Jumps as Well as Diffuses?—Lévy Processes and Lévy–Khintchine TheoryCompensated small jumps, Lévy triplets, original-space decomposition, stable attraction and jump limits.28Why Does Random Motion Produce Deterministic Equations? Semigroups, Generators and ResolventsExpected observations, generator domains, discounted occupation, Yosida construction and regular Markov paths.29When Small Probabilities Become Exponentially Small—Large DeviationsFrom all-heads records to exponential tilting, entropy, empirical measures and squared Brownian path energy.

Volume V · The Calculus of Random Motion

Chapters 30–40

30What Does It Mean to Know Before It Happens?Newly observed arrivals, optional and predictable information, regularization, Doob–Meyer and finite-variation compensation.31How Can We Integrate Along a Nondifferentiable Random Path?—Itô Integral and Quadratic VariationLeft sums and surviving squares, Itô isometry, predictable L2 extension, continuous martingale brackets and random energy clocks.

Volume VI · A World of Symmetry, Stationarity and Exchangeability

Chapters 41–47 · Planned · Text not yet written

Volume VII · From Random Points to Random Spaces

Chapters 48–53 · Planned · Text not yet written

Volume VIII · Random Trees, Continuous Geometry and New Probability Structures

Chapters 54–58 · Planned · Text not yet written

Topics and Mathematical Proofs

Browse the six topics and companion proofs by their chapter ownership.

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Sources and Versions
About this reading edition

All 31 chapters, advanced modules, supplementary proofs, references and writing guidance have full English working translations. Chinese manuscripts are protected; only explicitly authorized errata and cross-references are applied. Translations await semantic review and do not change the independent mathematical or historical review states.

Chapter 11 has been adopted by the author. Chapter 13 and its latest refinements have been adopted; Chapter 12 awaits author reading. Chapter 14 is a complete draft under mathematical verification and awaits formal author confirmation. Chapter 15 is a complete bilingual continuation draft awaiting author review; T-LP provides its analysis prerequisites. The published twelve-chapter edition has not been replaced. See the version page for the separate chapter states.