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an explainer on the mathematics of chance

Games · Updated 17 August 2026

Betting markets, prices and the overround

Add up the implied probabilities of every outcome in a market. The amount by which the total exceeds certainty is the margin.

Two priced bars whose combined length passes a dashed hundred per cent line
Two prices of 10/11 each imply 52.4 per cent. Together they imply 104.8 per cent of a market that can only sum to 100.

From a price to a probability

Fractional odds of a to b imply a probability of b/(a+b). A price of 10/11 -- win ten for every eleven staked -- implies 11/21 = 0.5238, or 52.4 per cent. In decimal form the same price is 1.909, and the implied probability is one divided by that. These implied figures are not the compiler's honest belief about the event; they are the belief with a margin added.

Summing the book

A market must resolve to exactly one outcome, so genuine probabilities sum to 1. Implied probabilities from quoted prices sum to more. Two sides priced at 10/11 sum to 1.0476: the book is 104.8 per cent, and the overround is 4.8 percentage points. Expressed as a share of the total staked if a bettor backed every outcome in proportion, the margin is 0.0476/1.0476 = 4.5 per cent. That is the operator's theoretical hold.

Reading the margin in a few book shapes
MarketPricesImplied sumMargin
Two-way, tight1.95 / 1.950.5128 + 0.5128 = 1.02562.5%
Two-way, typical1.909 / 1.9090.5238 + 0.5238 = 1.04764.5%
Three-way2.30 / 3.40 / 3.200.4348+0.2941+0.3125 = 1.04144.0%
Wide outright, 10 runnerssum of 10 prices1.200016.7%

Why margins widen with the field

The margin is spread across every outcome quoted, so a market with many possible results usually carries a larger total overround than a two-way market. A ten-runner outright book at 120 per cent implies a margin of 16.7 per cent of stakes; the same operator may run a 102.5 per cent book on a two-way market in the same event. The difference is not a judgement about the sport, but about how much uncertainty is being priced and how much competition exists on that market.

Prices move with money

A price is a commercial instrument, not a forecast. Compilers open at a position informed by models and form, then adjust as stakes arrive, aiming to hold a margin across whatever outcome occurs rather than to be right about any single one. This is why a heavily backed selection shortens even when nothing about the event has changed, and why closing prices tend to be better estimates than opening ones: they carry the aggregate of what everyone was willing to stake.

The margin is structural

Because the margin is embedded in every price, it applies to every bet placed at those prices regardless of outcome. A bettor whose judgement is exactly as good as the market's loses at the rate of the margin, repeatedly and without any bad luck being involved. Beating a priced market requires being better than the price by more than the margin, consistently, which is a far stronger claim than being right more often than not.

What this does not tell you

These conversions describe the arithmetic of quoted prices only. No market, operator or product is named or assessed here, and nothing on this page is a suggestion to bet.

How these guides are worked out

Every figure on this site is derived from the stated rules of the game it describes, using elementary probability, and the arithmetic is printed beside the result so that a reader can check it. No gambling company, product or offer is named, rated or linked anywhere on the site, and nothing here is certified or assessed. These guides explain how the games work and why the arithmetic favours whoever sets the terms; they do not explain how to play, and they treat gambling as a subject rather than an activity to take up.

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