An arbitrage, or surebet, is a set of bets covering every outcome of an event at prices good enough that you profit whatever happens. It exists because bookmakers disagree. One book prices a tennis match at 2.10 / 1.75, another at 1.80 / 2.10; take the 2.10 at each and you have backed both players at prices whose implied probabilities add up to less than 100%. This guide covers the arithmetic, the tools, and the things that go wrong.
The test
Add the implied probabilities of the best price on each outcome. If the sum is below 100%, you have an arbitrage; the return on your total stake is one divided by the sum, minus one.
S = Σ 1 / odds_i (best price per outcome)
arbitrage if S < 1
ROI = 1 / S − 1
Two-way, 2.10 and 2.10
1/2.10 + 1/2.10 = 0.4762 + 0.4762 = 0.9524. That is below 1, so it is an arbitrage, with a return of 1/0.9524 − 1 = 5.00%. On a total stake of 100 you make 5.00 whichever side wins. See it in the
arbitrage calculator
.
Prices this generous are rare on the same market at the same time at two bookmakers you can both use. Real arbitrages are usually 1–3%, and most of the work is finding them before they close.
Sizing the stakes
To profit the same amount whatever happens, each stake must be proportional to the implied probability of its outcome. Divide each outcome's implied probability by the sum S and multiply by the total stake.
stake_i = total × (1 / odds_i) / SThree-way, 3.00 / 3.20 / 3.50, total 1,000
Implied: 0.3333 + 0.3125 + 0.2857 = 0.9315. Return: 1/0.9315 − 1 = 7.35%. Stakes: 1,000 × 0.3333/0.9315 = 357.83; 1,000 × 0.3125/0.9315 = 335.46; 1,000 × 0.2857/0.9315 = 306.71. Check: 357.83 × 3.00 = 1,073.5; 335.46 × 3.20 = 1,073.5; 306.71 × 3.50 = 1,073.5. Every outcome returns 1,073.5 on 1,000 staked. Try it in the
calculator
.
When one leg is already placed
Often you have placed one bet and want to cover it later at a better price than expected. Fix that stake and solve for the others: each remaining stake is the fixed stake times the ratio of implied probabilities. The calculator's fixed leg mode does this and reports the resulting total.
Fixed leg
You have 200 on a team at 2.30. The other side is now available at 1.95. Implied: 0.4348 and 0.5128, sum 0.9476 — an arbitrage of 5.5%. The covering stake is 200 × 0.5128/0.4348 = 235.9, and the total becomes 435.9, returning 460 on either side.
Exchanges, commission and lay bets
Betting exchanges add two twists. Their prices carry commission, and they let you lay.
Commission is charged on net winnings, so a price on an exchange is worth less than it shows. At 5% commission, 2.10 becomes 1 + 1.10 × 0.95 = 2.045. Use the effective price in the arbitrage test, not the displayed one. The commission calculator converts any price.
Laying means betting that an outcome will not happen. A lay at odds L against a backer's stake B risks a liability of B × (L − 1) and wins B if the outcome loses. Mathematically it is a back bet on "not this outcome" at odds L / (L − 1), staked with the liability. That lets the classic hedge — back at a bookmaker, lay the same selection on the exchange — be solved with the same equations.
Back 2.20 at a bookmaker, lay 2.10 on an exchange at 5%
The lay is equivalent to backing "not X" at 2.10/1.10 = 1.909, which after 5% commission is 1 + 0.909 × 0.95 = 1.864. Implied: 1/2.20 + 1/1.864 = 0.4545 + 0.5366 = 0.9911. That is an arbitrage of 0.9%. On 100 total: back 45.9 at the bookmaker, put 54.1 of liability on the exchange (a backer's stake of 49.2). Either result pays 0.9. The
calculator
shows both stakes and the backer's stake to match.
When commission is charged on turnover rather than winnings, there is no single effective price, and the calculator solves the equal-profit equations directly instead of using the formula above. The result is the same kind of stake plan; the arithmetic underneath is a small linear system.
Rounding
Bookmakers accept stakes in whole units, and unusual stakes like 357.83 are also a flag that you are arbitraging. Rounding to 360 / 335 / 305 changes each outcome's profit. The calculator's rounding mode rounds every stake to the increment you choose and recomputes the profit on every outcome; it reports the worst and best case and whether the arbitrage survives.
The general rule: a thin arbitrage does not survive rounding. Below about 0.5% the profit is smaller than what rounding costs on modest stakes, which is why the calculator warns at that level.
Finding arbitrages
- Compare the same market across many books. Odds comparison sites and screens make this fast; by hand it is slow, and arbitrages close in minutes.
- Watch for slow books. Arbitrages appear when one book moves and another lags: after news, at line changes, around the start of an event.
- Use exchanges as the second side. Exchange prices are set by traders and often sit a tick above the bookmaker on the side the public is not backing.
- Check the terms are identical. Two prices only form an arbitrage if the bets settle the same way. Different rules for retirements (tennis), overtime (hockey), pushes (whole-number spreads) or dead heats (racing) can leave you with one leg paid and the other void.
What goes wrong
The price moves. The most common failure. You place the first leg; by the time you place the second, its price has shortened and the arbitrage is gone. Place the leg more likely to move first (usually the exchange or the sharper book), and re-check the second before confirming.
A leg is limited or rejected. Books limit stakes on arbitrage-prone markets, and some reject bets after a delay. If you can only get part of the stake down on one leg, fixed-leg mode will size the rest around what you got.
A leg is voided. A postponed match, a withdrawn player, a palpable-error void: one side paid out or refunded, the other left open. Read the rules before, not after.
Account restrictions. Bookmakers restrict or close accounts that only bet the bookmaker side of arbitrages. Rounding stakes, mixing in ordinary bets, and spreading action across books all reduce the signal, but none eliminate it. Treat any arbitrage strategy at bookmakers as one with a limited lifespan per account.
Exchange liquidity. A lay at 2.10 for 500 may only be available for 50. Check the amount available, not just the price.
Is it worth doing?
At 1–3% per arbitrage with capital tied up for hours or days, arbitrage is a low-margin, high-effort activity that depends on speed and on keeping accounts open. It teaches the maths of prices better than anything else, and the equal-profit stake calculation is the same one used for hedging a position you actually want to reduce. As a standalone strategy, be realistic about the returns after the failures above, and never chase a leg that has already moved.
Middles and scalps: the cousins of arbitrage
Two positions look like arbitrages and are often confused with them. A middle is a pair of bets on a spread or total at different lines: over 2.5 at one book and under 3.0 at another, or -3.5 at one and +4.5 at another. If the result lands in the gap, both bets win; outside it, one wins and one loses and you are down the margin on the loser. A middle is not a guaranteed profit; it is a cheap bet on the gap with the two books paying most of the cost. Size it like any bet on an outcome you have priced, not with the equal-profit formula, because there are three outcomes (low, gap, high), not two.
A scalp is a middle where the lines are identical and the prices differ enough that both sides together return more than the stake: the ordinary two-way arbitrage on a spread market. The push risk is the catch. On a whole-number line the game can land exactly on the number, refunding one leg while the other stands; the position then returns the stake on that leg plus the result of the other, which can be a loss. Half-point lines have no push and make clean arbitrages; whole numbers need a third outcome in the plan, and the calculator cannot promise equal profit across a push unless the pushed leg is modelled as a leg that returns its stake.
Capital, time and expectations
An arbitrage ties up the whole stake until the event settles. At 2% per position, 1,000 of capital deployed twice a week earns about 40 a week before failures. That is the scale: modest returns on capital that must sit idle at several books, with accounts that eventually restrict. The people who make it pay run many accounts, automate the search, and treat each account as a depleting asset. For everyone else the value of learning the maths is elsewhere: knowing what a price is worth, sizing a hedge on a bet you actually want to reduce, and recognising when two books disagree enough to matter. The calculator does that arithmetic in a second; the judgement about whether to act on it is the part that takes years.
Checklist
- Same market, same rules at every book.
- Best price per outcome, converted to effective odds after commission.
- Implied probabilities sum below 100%.
- Stakes in proportion to implied probability; round and re-check.
- Faster-moving leg first; confirm the second price before placing.
- Record everything.