In gambling parlance, making a book is the practice of laying bets on the various possible outcomes of a single event. The phrase originates from the practice of recording such wagers in a hard-bound ledger (the "book") and gives the English language the term bookmaker for the person laying the bets and thus "making the book". The mathematical basis of bookmaking is the management of risk through price adjustment.

Traditional models assume a static market where odds are set based on estimated probabilities and a fixed margin. A bookmaker strives to accept bets on the outcome of an event in the right proportions in order to make a profit regardless of which outcome prevails. See Dutch book theorems.

This is achieved primarily by adjusting what are determined to be the true odds of the various outcomes of an event in a downward fashion; the bookmaker pays out using actual odds that are lower than the "fair" value, thus ensuring a profit. While these fixed models provide the basis for betting arithmetic, modern dynamic bookmaking (often utilized in online environments) utilizes real-time data feeds and risk-management algorithms to adjust odds continuously as wagers are placed to manage the bookmaker's exposure.

The odds quoted for a particular event may be fixed but are more likely to fluctuate in order to take account of the size of wagers placed by the bettors in the run-up to the actual event (e.g., a horse race). This article explains the mathematics of making a book in the (simpler) case of the former event. For the second method, see parimutuel betting.

The relationship between fractional and decimal odds is a fundamental component of bookmaking arithmetic. Fractional odds are expressed as ${\displaystyle a-b}$ (or ${\displaystyle a/b}$), where a winning bettor receives their stake back plus ${\displaystyle a}$ units for every ${\displaystyle b}$ units wagered. Decimal odds (${\displaystyle D}$) represent a single value greater than 1, indicating the total payout per unit bet.

The formula to convert fractional odds to decimal odds is:

For example, a wager of £40 at 6 − 4 (fractional) results in a payout of ${\displaystyle {\text{£}}40+{\text{£}}60={\text{£}}100}$. The equivalent decimal odds are 2.5 (${\displaystyle 40\times 2.5=100}$). Conversely, fractional odds of ${\displaystyle a-1}$ (where ${\displaystyle b=1}$) are obtained from decimal odds via ${\displaystyle a=D-1}$.

Implied probabilities represent the theoretical likelihood of an outcome as suggested by the quoted odds. Fractional odds of ${\displaystyle a-b}$ correspond to an implied probability (${\displaystyle P}$) of: