\documentclass{article}
\usepackage[margin=2.7cm]{geometry}
\usepackage{oddsmath}
\usepackage{booktabs}
\usepackage[hidelinks]{hyperref}
\title{The \textsf{oddsmath} package\\\large Odds conversion and implied-probability macros}
\author{Edward Glush\\Bet Better Pty Ltd, Australia\\\url{https://betbetter.world}}
\date{2026/10/01 \quad v1.0}
\begin{document}
\maketitle

\section{What it does}
\textsf{oddsmath} provides expandable macros for documents about wagering
markets: converting between decimal, American and fractional odds, turning
prices into implied probabilities, removing the bookmaker's margin
(overround) to obtain fair probabilities, and computing expected value and
Kelly stake fractions. All arithmetic uses \textsf{xfp}'s \verb|\fpeval|,
so every macro expands to a plain number that can be used in tables and
further calculation.

It was written while preparing teaching materials and working papers on
market calibration at Bet Better
(\url{https://betbetter.world/studies/market-calibration}); worked
examples below use real market shapes from the free data feeds documented
at \url{https://betbetter.world/api}.

\section{Macros}
\begin{tabular}{@{}lll@{}}
\toprule
Macro & Example & Result \\
\midrule
\verb|\DecToProb{2.50}| & implied probability & \DecToProb{2.50} \\
\verb|\DecToPct{2.50}| & implied percent & \DecToPct{2.50} \\
\verb|\ProbToDec{0.40}| & fair decimal odds & \ProbToDec{0.40} \\
\verb|\DecToAmerican{2.50}| & American odds & \DecToAmerican{2.50} \\
\verb|\AmericanToDec{-150}| & decimal odds & \AmericanToDec{-150} \\
\verb|\FracToDec{5}{2}| & decimal from fractional & \FracToDec{5}{2} \\
\verb|\Overround{1.91}{1.91}| & two-way overround \% & \Overround{1.91}{1.91} \\
\verb|\OverroundIII{2.6}{3.3}{2.9}| & three-way overround \% & \OverroundIII{2.6}{3.3}{2.9} \\
\verb|\FairProb{1.91}{1.91}| & de-margined probability & \FairProb{1.91}{1.91} \\
\verb|\FairProbIII{2.6}{3.3}{2.9}| & fair probability, 3-way & \FairProbIII{2.6}{3.3}{2.9} \\
\verb|\ExpectedValue{2.50}{0.44}| & EV of 1-unit stake & \ExpectedValue{2.50}{0.44} \\
\verb|\KellyFraction{2.50}{0.44}| & Kelly fraction & \KellyFraction{2.50}{0.44} \\
\bottomrule
\end{tabular}

\section{Notes}
A typical two-way market priced 1.91/1.91 carries an overround of
\Overround{1.91}{1.91}\,\%: the two implied probabilities sum to more than
100\,\%, and the excess is the bookmaker's margin. Dividing each implied
probability by their sum (\verb|\FairProb|) removes that margin under the
proportional model. Empirically, fair probabilities computed this way track
observed outcome rates closely; see the open study of 18{,}238 settled
markets at \url{https://betbetter.world/studies/market-calibration} and
the margin survey at \url{https://betbetter.world/studies/margin-index}.

\section{License}
Released under the LaTeX Project Public License v1.3c or later.
Maintained by Edward Glush (\texttt{edward@betbetter.world}).
\end{document}