How Bookmakers Price Markets
From probability to price: what the overround is, how to measure it and why prices differ.
Every price on a bookmaker's board starts life as an estimate of how likely something is. What the customer sees is that estimate after it has been shaded. Add up the implied probabilities of every outcome in a market and the total comes to more than 100%; that excess is the overround, and it is the bookmaker's expected profit. That is the whole trick, and it is not hidden. It is arithmetic anyone can do at the counter.
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From probability to price
For fractional odds of a/b, the implied probability is b/(a+b). For decimal odds D, it is simply 1/D. Those two formulas, set out in Wikipedia's article on the mathematics of bookmaking, are the only tools needed to read a betting market honestly.
Convert every price in a market to an implied probability and add them up: anything above 100 per cent is the margin.
| Market | Prices | Implied probabilities | Total |
|---|---|---|---|
| Two-way, with margin | 10/11 and 10/11 | 52.4% + 52.4% | 104.8% |
| Two-way, no margin | Evens and evens | 50% + 50% | 100% |
| Three-way football | 6/4, 9/4, 2/1 | 40% + 30.8% + 33.3% | 104.1% |
The margin is usually widest in markets with many outcomes, such as a first goalscorer or an outright, and narrowest where competition forces prices out.
Once every selection is expressed as a percentage, the shading becomes visible. puts it in plainer trade language: overround is the practice of factoring in a profit margin, and it is best displayed as a percentage. Both refer to the same thing that the academic literature calls vigorish, or just the vig.
The distinction that trips people up: the margin is not a statement about the event. It is the gap created by summing prices across outcomes. A bookmaker can be dead right about a team's chances and still be offering a price that loses money over time for anyone taking it.
Measuring the overround
Take every price in a market. The amount by which the book exceeds 100% is the overround. That paper also gives the consequence in the punter's terms. The overround determines the expected long-run return, which is 1/π. Wider book, worse return. There is no mechanism by which a higher margin benefits the person placing the bet. One caveat worth holding onto. The overround is not a fixed percentage stamped across a bookmaker's entire offering. It varies by market.
Worked examples
A two-way market with both sides quoted at 10/11 works out at 1.909 in decimal, which is 52.4% each. The two sides add to 104.8%, giving an overround of 4.8%. Strip the margin out entirely and the same event prices at evens, 2.00 decimal, 50% a side. That 4.8 points is the difference between a fair coin and a commercial book. Now a three-way football market priced at 6/4, 9/4 and 2/1. The implied probabilities are 40%, 30.8% and 33.3%. Treat both as snapshots, not templates. No bookmaker runs every market at 4.1%.
Why prices differ across firms
Two firms can hold an identical view of a match and still post different odds, because the margin they apply and the way they distribute it across outcomes are separate decisions. A book can be loaded more heavily against the favourite or spread evenly. The sum tells you the total take; it does not tell you which selection is carrying it. This is also why comparing a single price between two firms is close to meaningless without doing the full conversion. One bookmaker's 2/1 might sit inside a 103% book, another's inside a 107% one.
Where the numbers come from
The normalisation method is standard in the research literature. A University of Reading paper on betting markets for English Premier League results uses πᵢ = (1/dᵢ) / Σ(1/dᵢ), dividing each inverse decimal price by the sum of them all to recover probabilities from quoted odds. Which is the point. The fair-probability estimate is only the starting input; the overround is what converts it into a tradable price, and that is why the same fixture carries different numbers in different shops.