Chapter 2 Further Reading: Probability and Odds

Probability Textbooks

1. Introduction to Probability by Joseph K. Blitzstein and Jessica Hwang (2nd Edition, CRC Press, 2019) The best modern introduction to probability for anyone approaching sports betting analytically. Blitzstein's treatment of conditional probability, Bayes' theorem, and the Law of Large Numbers is exceptionally clear and grounded in real-world examples. The chapter on the Gambler's Ruin problem is directly relevant to bankroll management concepts explored later in this textbook. The freely available Harvard Stat 110 lecture series that accompanies this book is equally valuable.

2. Probability and Statistics for Engineering and the Sciences by Jay Devore (9th Edition, Cengage, 2016) A standard university-level textbook that provides rigorous coverage of probability axioms, combinatorics, random variables, and expected value. While not betting-specific, Chapters 1 through 5 provide the mathematical backbone needed to understand implied probability, vig removal, and the probabilistic reasoning that underpins every concept in this course. Recommended for readers who want to formalize their understanding.

3. The Art of Statistics: How to Learn from Data by David Spiegelhalter (Basic Books, 2019) An accessible and engaging introduction to statistical thinking by one of the world's leading statisticians. Spiegelhalter emphasizes the interpretation of probability and the communication of uncertainty, which are directly applicable to evaluating betting odds and understanding what probability estimates actually mean. Particularly useful for readers who prefer a less formula-heavy, more conceptual approach.

4. Probability Theory: The Logic of Science by E.T. Jaynes (Cambridge University Press, 2003) An advanced treatment of Bayesian probability that argues probability should be understood as a measure of rational belief rather than long-run frequency. This philosophical perspective is deeply relevant to sports betting, where probabilities are often subjective estimates rather than frequencies derived from repeated trials. Not for beginners, but essential reading for those who want to understand the foundations of Bayesian reasoning in betting.

Odds and Betting Mathematics

5. Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets by Catalin Barboianu (Infarom, 2006) A comprehensive mathematical treatment of gambling across all its forms. The chapters on sports betting odds and probability extraction are particularly relevant, covering the derivation of implied probabilities, the mathematics of overround, and the formal structure of different odds formats. One of the few books that treats gambling mathematics with full academic rigor.

6. Fixed Odds Sports Betting: Statistical Forecasting and Risk Management by Joseph Buchdahl (High Stakes Publishing, 2003) The definitive text on the mathematics of fixed-odds betting. Buchdahl covers odds conversion, margin analysis, and the statistical testing of betting systems with exceptional thoroughness. His treatment of how to evaluate whether a betting record reflects genuine skill versus luck is invaluable. Chapter 2 on probability and odds is the single most relevant external chapter to pair with this textbook's Chapter 2.

7. Squares and Sharps, Suckers and Sharks: The Science, Psychology, and Philosophy of Gambling by Joseph Buchdahl (Oldcastle Books, 2016) Buchdahl's follow-up work broadens the scope to include behavioral psychology, market efficiency, and the philosophical underpinnings of sports betting. The sections on how bookmakers set and adjust odds provide essential context for understanding what implied probabilities actually represent and how the vig functions as a business model.

8. Trading Bases: How a Wall Street Trader Made a Fortune Betting on Baseball by Joe Peta (Dutton, 2013) A former Wall Street trader applies financial modeling to baseball betting. While more narrative than technical, Peta's discussion of converting between different probability representations and his methodology for identifying value in moneylines offers practical illustration of the concepts in this chapter. Accessible to readers without a strong math background.

Academic Papers on Betting Markets

9. Shin, H.S. (1991). "Optimal Betting Odds Against Insider Traders." The Economic Journal, 101(408), 1179-1185. The foundational paper on extracting true probabilities from bookmaker odds by modeling the presence of informed bettors. Shin's method provides an alternative to the simple multiplicative vig-removal approach and is widely used in academic research on betting markets. Essential reading for anyone interested in sophisticated approaches to probability extraction from odds.

10. Shin, H.S. (1993). "Measuring the Incidence of Insider Trading in a Market for State-Contingent Claims." The Economic Journal, 103(420), 1141-1153. Shin's extension of his earlier work, providing a refined method for estimating the proportion of informed trading in a betting market. This paper formalized the "Shin probability" approach that has become a standard tool in the academic literature on sports betting efficiency.

11. Levitt, Steven D. (2004). "Why Are Gambling Markets Organised So Differently from Financial Markets?" The Economic Journal, 114(495), 223-246. The Freakonomics author examines how sportsbooks set lines and why the structure of betting markets differs from stock markets. Levitt demonstrates that bookmakers do not simply balance their books but instead exploit bettors' known biases. This paper provides crucial context for understanding what odds actually represent: not just probability plus vig, but probability plus vig plus exploitation of systematic bettor errors.

12. Boulier, Bryan L. and Stekler, Herman O. (2003). "Predicting the Outcomes of National Football League Games." International Journal of Forecasting, 19(2), 257-270. An empirical study of the accuracy of NFL betting lines as forecasts. The authors find that closing lines are remarkably well-calibrated, supporting the efficient market hypothesis for sports betting. Directly relevant to the calibration concepts discussed in this chapter and provides empirical evidence for why beating the closing line is the gold standard of betting skill.

13. Forrest, David, Goddard, John, and Simmons, Robert (2005). "Odds-setters as Forecasters: The Case of English Football." International Journal of Forecasting, 21(3), 551-564. An analysis of how well bookmaker odds predict outcomes in English soccer. The paper examines the overround structure of three-way markets (home/draw/away) and finds that bookmakers are skilled forecasters but that their margins are not applied uniformly across outcomes. This finding has implications for which vig-removal method is most appropriate.

Online Resources and Data

14. Pinnacle Sports Blog and Betting Resources (pinnacle.com/betting-resources) Pinnacle, known in the industry for having the sharpest (lowest-margin) lines, publishes an extensive library of free articles on betting mathematics, probability, and odds analysis. Their series on understanding value, odds conversion, and margin calculation is written for practitioners and directly complements this chapter. Pinnacle's odds are often used as the benchmark "true" line in academic research.

15. Football-Data.co.uk (football-data.co.uk) The most comprehensive free archive of historical soccer betting odds, covering major European leagues with closing odds from dozens of bookmakers going back to the 1990s. Invaluable for the data analysis exercises in this chapter and for calibration studies. The site also provides detailed documentation of its data format and odds sources.

16. Odds Portal (oddsportal.com) A multi-sport odds comparison platform that tracks live and historical odds from hundreds of sportsbooks worldwide. Useful for comparing margins across bookmakers, studying odds movement, and identifying the best available lines. The site displays odds in all major formats (American, decimal, fractional) and is an excellent practical tool for understanding how the same probability is expressed differently across markets.

17. The Power Rank (thepowerrank.com) by Ed Feng A data science-driven sports analytics site that explains the statistical and probabilistic methods behind sports prediction models. Ed Feng, who holds a PhD in chemical engineering, writes clearly about converting model outputs into betting probabilities and evaluating their calibration. The site bridges the gap between academic probability theory and practical sports betting application.

18. Beating the Bookies with Their Own Numbers by Lisandro Kaunitz, Shenjun Zhong, and Javier Kreiner (2017, arXiv:1710.02824) A widely discussed working paper that demonstrates a systematic strategy for exploiting inefficiencies between different sportsbooks' odds. The paper's methodology relies heavily on odds conversion, implied probability calculation, and margin analysis, making it an excellent case study for the techniques covered in this chapter. The authors found statistically significant positive returns by consistently betting where their model identified value relative to the market consensus.

How to Use This Reading List

If you are new to probability: Start with Blitzstein and Hwang (#1) or Spiegelhalter (#3) to build a solid conceptual foundation before diving into betting-specific material.

If you want practical betting mathematics: Buchdahl's Fixed Odds Sports Betting (#6) is the single most valuable companion text to this chapter.

If you want to go deeper into theory: Read the Shin papers (#9, #10) and Jaynes (#4) for advanced probability extraction and Bayesian reasoning.

If you want to work with real data: Start with Football-Data.co.uk (#15) and Odds Portal (#16) and attempt the analysis exercises in this chapter with live data.

If you want academic rigor: The Levitt (#11), Boulier and Stekler (#12), and Forrest et al. (#13) papers provide the empirical evidence base for many claims made in this chapter.