Reference
Probabilities vs Bookmaker Odds: The Margin, Explained
Illustrative distributions used below. Every bar is an example: the shares describe no real fixture, and no LeagueQuant forecast lies behind them.
- Home 46.08%
- Draw 28.46%
- Away 25.46%
- Home 38.42%
- Draw 29.56%
- Away 32.02%
Two different objects
A probability and a price look alike at first glance, and that resemblance is where the confusion begins. They are different objects, made for different purposes, and this page is about the difference.
A probability is a share of chance. A three-way football market has three outcomes — home win, draw, away win — and a distribution gives each one a share, the three shares summing to exactly 100. A distribution is a statement about how the remaining uncertainty divides.
A price is a commercial quantity, quoted by a firm that sells positions on all three outcomes. It must bring in enough money over time to pay for staff, data, systems and the ordinary cost of being wrong in an unlucky week. A price carries a job a probability does not: keeping the firm that sets it in business.
The two look comparable because a decimal price turns into a probability-shaped number with one division; the rest of this page is about that conversion and its systematic difference from a true distribution.
Implied probability
For decimal prices the conversion is one division:
implied probability = 1 ÷ decimal price
As a percentage, the result is the share the price implies for its outcome. Take a three-way market priced at 2.10 for the home win, 3.40 for the draw and 3.80 for the away win:
| Outcome | Decimal price | Implied share |
|---|---|---|
| Home | 2.10 | 47.62% |
| Draw | 3.40 | 29.41% |
| Away | 3.80 | 26.32% |
Each row is one division: 1 ÷ 2.10 = 0.476190, or 47.62%; 1 ÷ 3.40 = 0.294118, or 29.41%; 1 ÷ 3.80 = 0.263158, or 26.32%. Fractional prices translate directly: fractional 11/10 is the same price as decimal 2.10.
Adding the column, the shares sum to 103.35% — at full precision, 103.3466%. A probability distribution cannot sum to 103.35%. Something systematic is happening.
Why the implied shares sum to more than 100
The reason is structural, not accidental.
The firm sells all three outcomes. Exactly one of them occurs, and the firm pays out on that one. Its income is the money taken in across the market; its outgoings are the payouts on the outcome that took place. Prices must therefore be set so that, on average, the money taken in exceeds the money paid out. If the three implied shares summed to exactly 100, the price-setter would have no gross margin at all — the arithmetic would leave nothing over for the cost of running the market.
The fingerprint of that arrangement is that each outcome is priced slightly short of its share of 100, and the three shortfalls together produce the excess. The excess above 100 is the overround. Here the shares sum to 103.35%, so the overround is 3.35 percentage points.
This is an ordinary commercial arrangement, not a trick: a firm that runs a priced market, like any other seller, prices above cost. Every market of this kind behaves this way, and nothing is hidden — the prices are public, the conversion is one division per price, and the excess falls out for anyone who looks.
Overround and margin are not the same number
The two are routinely conflated: they are close, related, and not equal.
The overround is the excess above 100, in percentage points: 3.35 points here.
The margin as a share of the book is the fraction of the money taken in that the firm keeps, before its own costs, when positions are taken in proportion to the prices:
margin = 1 − (1 ÷ booksum)
where booksum is the sum of the implied shares. For this market, 1 − (1 ÷ 1.033466) = 3.24%.
The numbers differ because they take the excess against different baselines: the overround against 100, the margin against the whole inflated sum. The money that changes hands across a balanced market is proportional to 103.35, so the excess of 3.3466 is 3.3466 parts in 103.3466 — 3.24%, not 3.35%.
The gap widens as the book gets fatter. A two-way market with both sides at 1.90 shows it plainly:
- 1 ÷ 1.90 = 0.526316, so each side implies 52.63%
- the shares sum to 105.26%
- the overround is 5.26 percentage points
- the margin is 1 − (1 ÷ 1.052632) = 5.00%
The margin there is exactly 5.00%: the booksum is exactly 20/19, and 1 − 19/20 = 1/20. At a 3.35-point overround the margin is 3.24% — a gap of 0.11 points; at a 5.26-point overround, 5.00% — a gap of 0.26 points. Neither figure is wrong; they answer different questions. Quoting the overround and calling it the margin is the common error, and it always overstates the margin.
Normalising
Dividing each implied share by the booksum strips the margin out proportionally and leaves shares that sum to 100. Here each share is divided by 1.033466:
- Home: 46.08%
- Draw: 28.46%
- Away: 25.46%
The sum is 100.00% — exactly the first bar at the top of this page. The second bar shows the same procedure on a tighter market — 2.50, 3.25, 3.00 — with implied shares of 40.00%, 30.77% and 33.33%, a booksum of 104.10, a margin of 3.94%, and normalised shares of 38.42%, 29.56% and 32.02%.
The honest caveat: proportional normalising is a convention, not a recovery of the price-setter’s true opinion. It assumes the margin sits on every outcome in proportion to its share — that the padding is even. Nothing requires a real firm to pad evenly: it may shade a price where its own opinion is weakest, where it expects demand mostly on one side, or where rivals’ prices constrain it. Real books are not padded evenly, so the normalised shares are an approximation, and the size of the error cannot be known from the prices alone. The prices say how much margin there is in total; they do not say where it was put.
Reading a distribution
A published probability distribution and a normalised price are two estimates of the same uncertain thing, made by different processes for different purposes. One may come from a model of matches; the other from a market in prices. Comparing the two is legitimate arithmetic: convert, normalise, subtract, and the differences are what they are — numbers with signs and sizes.
What the arithmetic on this page does not support is a conclusion that one estimate is wrong, or that a difference between them can be exploited. Those conclusions need premises this page does not have: how each estimate was built, and how the differences behave across many matches. The arithmetic converts and compares. It does not rank, and it recommends nothing.
The record so far
LeagueQuant publishes no forecasts yet and holds no accuracy record. Live forecasts will appear here in the future: each will be published with a hash recorded before kick-off, and each will be evaluated automatically after the match, with misses included in the record alongside everything else. Until then, this section publishes method and reference material only.