Sports Betting Math Glossary

The excess m=sum(1/Oi)-1 for a complete mutually exclusive market. It measures quoted overround, not realized hold or an individual selection’s expected loss. Under proportional probabilities, expected loss per unit stake is m/(1+m). Positive overround alone does not make every selection negative EV.

The reciprocal of decimal odds, 1/O. It is a break-even probability for that price, not an independently established outcome probability. Across a complete mutually exclusive market, the sum may exceed, equal or fall below one.

For a specified probability p>0, fair decimal odds are 1/p and give EV=pO-1=0. When p is estimated or obtained by de-vigging, the quoted fair odds are conditional on that model; normalization does not establish the true probability.

The standard numerical format for betting odds utilized in continental Europe, South America, and Australia. Decimal odds represent the total payout returned per unit currency staked, including the initial capital investment.

A transformation from complete mutually exclusive market prices to a probability estimate under a specified margin-allocation model. Multiplicative, Additive, Power and Shin use different assumptions; normalization does not recover observed truth or establish forecast calibration.

The most widely utilized devigging approach, which operates under the assumption that the sportsbook distributes its margin in strict linear proportion to the implied probability of each selection. It is simple and robust; it ignores the favourite-longshot bias, yet on 2,490 Premier League matches it was as accurate as Shin (Brier 0.5676).

A margin model introduced by Hyun Song Shin. Its assumptions about informed betting imply a nonlinear normalization equation; a feasible root can be found numerically, including by bisection. The parameter is model-based, not a measured insider share without validation.

Power normalizes reciprocal prices q_i by solving sum(q_i^k)=1. With 0<q_i<1, k>1 for positive overround, k=1 for a unit sum and k<1 for underround. The transformation is a model assumption, not proof of the true probability or a measured bias.

The last recorded pre-event quote under a specified source, timestamp and settlement rule. A liquid closing market can be a useful comparison benchmark, but it does not contain all information or directly reveal true probabilities.

A ubiquitous empirical anomaly in sports betting and financial markets where bettors systematically overvalue longshots (underdogs) and undervalue heavy favourites. Bookmakers exploit this behavioral skew by embedding disproportionately higher profit margins into high-odds selections.

The expected net return under specified probabilities and settlement. For a binary stake with decimal odds O and win probability p, EV per unit is pO-1. Negative expectation describes an average, not a guarantee of any finite sequence or individual outcome.

A positive expected return under specified probabilities, pO-1>0 for a binary stake. A positive estimated EV is a model conclusion; probability error, dependence, execution and stake sizing can still produce losses.

A price comparison against a specified closing benchmark. No-vig CLV uses a complete closing market and a named normalization model. It equals EV under the chosen closing probability estimate, not necessarily under the true probability, and is observable only after closing.

Expected net return per unit stake under a stated probability estimate: pO-1. Calling a price better than a model fair price identifies an estimated edge, not independently verified profit.

The finite second central moment E[(R-E[R])²] of returns. It describes dispersion, not a fixed probability of a particular losing streak. Streak probabilities also depend on the win probability, dependence and observation horizon.

A planned number of observations for a specified precision or hypothesis test. The displayed normal approximation estimates the sample size for a confidence-interval half-width E under independent finite-variance returns. It is not a universal threshold proving skill. Power calculations also need a null, effect size, significance and dependence assumptions.

A mathematical formula derived by John L. Kelly Jr. (1956) that determines the optimal proportion of an investment bankroll to allocate to an opportunity with positive expected value to maximize the asymptotic long-term geometric compounding growth rate of wealth.

Stake the full optimum of expected log growth under the specified model. For a binary bet with known p and odds O>1, use f*=(pO-1)/(O-1), constrained by the allowed stake domain. Finite-horizon drawdown probabilities depend on the model and horizon; there is no universal 50% drawdown probability.

Stakes 50% of the exact Kelly fraction. In the small-stake quadratic approximation it captures about 75% of maximum log growth; the exact ratio depends on probability and odds. For one-bet simple returns, standard deviation halves and variance becomes one quarter. Finite-horizon drawdown needs a separate calculation.

Stakes 25% of the exact Kelly fraction. The small-stake quadratic approximation gives about 43.75% of maximum log growth, rather than a universal exact ratio. For one-bet simple returns, standard deviation becomes one quarter and variance one sixteenth. This does not guarantee protection from probability error or large drawdowns.

The probability of first reaching a specified bankroll threshold a within horizon N. State whether a means zero wealth, a fixed fraction of the initial bankroll or an operational minimum. Peak-to-trough drawdown is a separate event.

The segregated pool of investment capital strictly designated for sports betting transactions. To prevent risk of ruin, stakes must be dynamically recalculated based on current, not initial, bankroll values.

A static capital allocation strategy where a fixed monetary amount or fixed percentage of initial bankroll (e.g. 1.0% or $100) is wagered on every proposition, irrespective of differing mathematical edges. While simple, it is mathematically suboptimal compared to Kelly.

A discrete probability distribution expressing the likelihood of a given number of events occurring within a fixed interval of continuous time or space, parameterized by expected rate $\lambda$. Because football goals are rare and discrete, Poisson forms the foundational mathematical engine for football prediction.

A model estimate of the probability that a shot becomes a goal, aggregated across shots. Features and calibration differ by provider. Whether using xG improves a goal-rate model must be tested out of sample; no fixed percentage improvement follows from the definition.

The relative offensive efficiency parameter of a football team relative to league average. An attack rating of 1.25 indicates that the squad creates 25% more goals per game than an average league opponent.

The relative defensive vulnerability parameter of a football club. A defence rating below 1.00 denotes superior defensive resistance (e.g. beta = 0.80 indicates conceding 20% fewer goals than league average).

A two-dimensional joint probability grid (typically 6x6 or 7x7) computed by multiplying the independent marginal Poisson probabilities of the home and away teams for all scorelines from 0-0 to 6-6.

A correction to four low-score cells of an independent Poisson model. The fitted parameter rho changes 0:0, 1:0, 0:1 and 1:1; its sign and feasible domain matter. Rho is a model parameter, not the Pearson correlation coefficient of goals.

The premier total goals betting market in football. The probability of Under 2.5 is the exact sum of probabilities for scorelines 0-0, 1-0, 0-1, 1-1, 2-0, and 0-2; Over 2.5 is its mathematical complement (1 - P(Under 2.5)).

A derivative betting proposition on whether both teams score at least one goal in regular match play. Under Poisson independence, it simplifies to the product of both teams' probabilities of scoring at least once.

The foundational three-way betting market in football: 1 denotes a Home Win, X denotes a Draw, and 2 denotes an Away Win at the conclusion of 90 minutes plus stoppage time.

A combined bet requiring all legs to win. Under independence and proportional probabilities for complete component markets, the combined quoted overround is product(1+m_i)-1, while expected loss per unit stake is 1-product(1/(1+m_i)). These are different quantities.

An informal label for a price-forming sportsbook used as a benchmark. It does not imply a fixed margin, universal acceptance of winners or unrestricted limits. Assess the specific market, timestamp and account terms; an exchange has a different commission structure.

An informal label for a consumer-facing sportsbook, often contrasted with price-forming benchmarks. The label alone establishes neither its overround nor update speed or account restrictions; these require current market-specific evidence.