Implied probability: why bookmaker odds add up to more than 100%

Give a fictional football match prices of 2.00 for the home win, 3.50 for the draw and 4.00 for the away win. Exactly one result can occur within the covered match period, yet the percentages total 103.57%. There is no fourth outcome hiding in the remaining 3.57 points. These prices contain a commercial margin.

That calculation is useful when reading an odds screen. It does not establish the teams' actual chances or tell anyone how to make gambling profitable. All prices and amounts below are invented teaching examples, with no live match or operator offer attached.

Start with what the decimal price pays

A £10 cash single at 2.00 returns £20 on a win: the £10 stake and £10 profit. If it loses, nothing returns. Suppose an ordinary bet can only win or lose, its accepted price stays fixed, and there are no charges, deductions or refunds. A 50% win probability would make its expected net result zero.

The arithmetic is £10 × (0.5 × 2.00 - 1) = £0. That is an average over possible outcomes, not a promise that two bets produce one win and one loss. For a decimal price d and win probability p, the net expectation per unit staked is p × d - 1. Setting that to zero gives p = 1/d.

This is why the reciprocal is also called a break-even probability under those assumptions. A price of 3.00 implies 33.33%, rounded to two decimal places. It says which probability would balance that payout against losing the stake. It does not show that a model, trader or independent observer has measured the chance as one in three.

Smarkets' conversion guide supplies the formulas for decimal, fractional and American displays. Fractional 2/1 and American +200 express the same gross return as decimal 3.00, hence the same 33.33%. Decimal 1.50, fractional 1/2 and American -200 each imply 66.67%. For a negative American price, use its positive magnitude: 200/(200 + 100), not -200/(-200 + 100).

The format changes how the quote is written. It cannot remove a bookmaker's margin. Nor does a current screen establish what was accepted earlier; our guide to odds changing after placement separates those records.

One football market, more than a whole

Return to the invented match. The three selections cover a home win, draw or away win over the same period, so they are mutually exclusive and collectively exhaustive. Each settled match has one of those outcomes. Convert every price before adding them.

Using the unrounded fractions, 1/2 + 2/7 + 1/4 = 29/28. Subtract 1 and convert to a percentage: the overround is approximately 3.57%, or 3.57 percentage points above a 100% book. A single outcome's price cannot supply this total.

Smarkets' explanation of bookmaker margins also treats the whole market as the relevant comparison. Use prices from one provider at one time, covering the same result period. Combining a home price from yesterday with today's draw quote creates a different calculation. Omitting the draw from a three-way market leaves the calculation incomplete.

Overround does not report the operator's realised profit. That depends on stakes accepted and what gets paid after settlement. It is also different from the expected loss on one particular selection, which requires an assumed probability. At fictional prices of 1.91 on each side of an exhaustive two-outcome market, the overround is about 4.71%. If each outcome truly has a 50% chance, a one-unit wager has expected loss of 1 - (0.5 × 1.91) = 0.045 units, or 4.5% of its stake. Those percentages have different denominators.

A 100% total still needs an assumption

Dividing each raw percentage by their combined total rescales the football example to exactly 100%. The home share becomes (1/2)/(29/28) = 14/29, about 48.28%. The draw becomes 8/29, about 27.59%, and the away share 7/29, about 24.14%.

The displayed rounded figures sum to 100.01%; the exact fractions sum to 1. This is proportional normalization, sometimes described as removing the vig or making a no-vig book. It applies the same relative scaling to every raw probability.

The calculation has assumed how the margin should be allocated. It has not recovered information about a team's injuries or established that the bookmaker priced every outcome equally accurately. Calling the resulting 48.28% a verified home-win chance would go beyond the arithmetic.

In the abstract of their 2017 paper, Stephen Clarke, Stephanie Kovalchik and Martin Ingram distinguish several adjustments and identify a limitation of normalization: it does not account for favourite-longshot bias. That is a warning against treating one rescaling method as a neutral route to the truth, not a claim about the accuracy of a particular match price today.

Refunds and charges change the question

The simple break-even formula assumed that every settled wager either wins at the quoted price or loses the whole stake. A drawn Draw No Bet single returns the cash stake under the named rules in our DNB settlement explainer. There are then three relevant states, including a refund with zero profit.

If w is the chance of a win, l the chance of a loss and the remaining chance produces only a full stake refund, break-even requires w × (d - 1) = l. Rearranging gives w/(w + l) = 1/d. The reciprocal describes the win share conditional on a non-refunded outcome in this simplified model. It cannot, by itself, give the unconditional chance of the team winning the match.

Fees create another boundary. An exchange back price can imply one raw percentage while commission changes the net amount received. Our exchange commission report follows the charging base and account terms. A conversion that ignores those terms cannot establish the customer's break-even point after fees.

An odds-to-percentage converter can faithfully translate the number typed into it. Whether that number describes a complete market, an uncharged cash wager or a selection with a refund state remains a question about the underlying bet. Keep those conditions beside the percentage; the decimal alone cannot supply them.