What is a simple example of arbitrage betting?

Illustratively, Book A prices Player X at 2.08 and Book B prices Player Y at 2.02 in a two-player match. Implied probabilities sum to 0.9758, below 1. Staking 492.68 on X and 507.32 on Y returns 1,024.78 whichever player wins, a profit of 24.78 on 1,000, before costs, limits, and execution risk.

How do you calculate arbitrage stakes?

Convert each best price to an implied probability with 1 / o and sum them to get S. For a total stake T, stake T x (1 / o) / S on each outcome. Every outcome then returns T / S, and the profit is T / S minus T. If S is 1 or above, there is no arbitrage.

Why does a three-way arbitrage need all three outcomes?

Because the guarantee only holds if every possible result is covered. In football match result markets the draw is an outcome, so leaving it out means the position loses everything staked whenever the match is drawn. Any market used for arbitrage must be mutually exclusive and exhaustive across the legs, including identical lines and periods.

Does exchange commission affect an arbitrage?

Yes, and it often removes it. Commission on net winnings lowers the usable price to 1 + (o - 1) x (1 - c). In the illustrative case on this page, a gap that looks like 0.74 percent on raw prices becomes a sum of 1.0054 after a 5 percent commission, which is no arbitrage at all.

Arbitrage Betting Explained With Worked Examples

Arbitrage betting means backing every outcome of a market at different books when the best prices' implied probabilities sum below 1. Illustratively, 2.08 and 2.02 on a two-player match sum to 0.9758; staking 492.68 and 507.32 returns 1,024.78 either way. Price moves, commission, limits, and settlement rules often erase that margin.

The one formula behind every arbitrage

Every arbitrage reduces to one test and one stake rule. All odds below are decimal and illustrative, chosen to make the arithmetic visible, not observed prices.

The test. For each outcome, take the best available price o and compute its implied probability 1 / o. Add these up across every mutually exclusive outcome. Call the sum S. If S is below 1, an arbitrage exists before costs. The theoretical return on the total stake is 1 / S minus 1.

The stake rule. With a total stake T, put stake_i = T x (1 / o_i) / S on outcome i. Every outcome then returns the same amount, T / S. Because each stake is proportional to its implied probability, the bigger stake always goes on the shorter price.

Worked example 1: a two-way tennis market

Tennis match winner has two outcomes and no draw, which makes it the cleanest case.

Book A offers Player X at 2.08. Implied probability 1 / 2.08 = 0.4808.
Book B offers Player Y at 2.02. Implied probability 1 / 2.02 = 0.4950.
S = 0.4808 + 0.4950 = 0.9758. Below 1, so this is an arbitrage. Return = 1 / 0.9758 - 1, about 2.48 percent.

With T = 1,000:

Stake on X = 1,000 x 0.4808 / 0.9758 = 492.68 at Book A.
Stake on Y = 1,000 x 0.4950 / 0.9758 = 507.32 at Book B.

If X wins: 492.68 x 2.08 = 1,024.78. If Y wins: 507.32 x 2.02 = 1,024.78. Either way the return is 1,024.78 on 1,000 staked, a profit of 24.78. A gap this wide is unusually large; real two-way gaps are typically much thinner and close quickly.

Worked example 2: a three-way football market

Football match result has three outcomes: home, draw, away. Missing the draw is the classic error, so all three must be in the sum.

Book A: home at 2.45, implied 0.4082.
Book B: draw at 3.50, implied 0.2857.
Book C: away at 3.30, implied 0.3030.
S = 0.4082 + 0.2857 + 0.3030 = 0.9969. Return = 1 / 0.9969 - 1, about 0.31 percent.

With T = 1,000, the stakes are 409.43 on home, 286.60 on draw, and 303.97 on away. Each outcome returns 1,003.10, a profit of 3.10.

This is closer to what three-way gaps tend to look like: a fraction of a percent. Three books, three accounts, three bets, all placed before any price moves, for a few units of profit. That ratio of effort and exposure to return is why the arithmetic is the easy part.

What breaks it in practice

The formulas assume every price is still there, every bet is accepted in full, and every bet settles on the same rules. Each assumption fails regularly.

A price moves before the last leg

In example 2, suppose the home and away bets are placed and the draw drops from 3.50 to 3.30 before the third bet goes on. S becomes 0.4082 + 0.3030 + 0.3030 = 1.0142, so the arbitrage is gone. Covering the draw at 3.30 to equalise returns needs a stake of about 303.97, total outlay 1,017.37 for a return of 1,003.10: a guaranteed loss of about 14.27. Leaving the draw unbacked turns the position into an outright bet against the draw.

Rounding and stake limits

Books accept stakes in set increments and cap maximum stakes. Rounding example 1 to 493 and 507 gives returns of 1,025.44 and 1,024.14, still positive but no longer equal. When a book caps the stake on the leg that needs the most money, the whole position has to shrink to fit that cap.

Exchange commission

On an exchange, commission is taken from net winnings, so the usable price is 1 + (o - 1) x (1 - c). Say Book A offers 2.00 on X and an exchange shows 2.03 on Y, with an illustrative 5 percent commission. Raw S is 0.5000 + 0.4926 = 0.9926, which looks like a 0.74 percent arbitrage. The effective exchange price is 1 + 1.03 x 0.95 = 1.9785, so S is really 0.5000 + 0.5054 = 1.0054. No arbitrage.

Settlement rules differ

Books can settle the same market differently: retirements in tennis, overtime in totals, abandoned matches, player participation in props. If one leg is voided and the other stands, the hedge disappears and what remains is a single bet.

Account limits

Even when every leg executes, books watch for accounts that only take out-of-line prices and commonly reduce their stakes. The arithmetic does not price that in.

Checking an arbitrage from your own data

The same sum is a useful check on any odds dataset. Compute S per market from the best price per outcome across books. A value far below 1 is more often a stale quote, a mismatched market (a different line or period), or a missing outcome than a real opportunity. Store observation timestamps per book so a stale price can be identified as stale rather than read as a gap. Computed within one book, S minus 1 is that book's margin on the market.

This page explains arbitrage arithmetic with illustrative numbers and is not betting advice.

Frequently asked questions

What is a simple example of arbitrage betting?: Illustratively, Book A prices Player X at 2.08 and Book B prices Player Y at 2.02 in a two-player match. Implied probabilities sum to 0.9758, below 1. Staking 492.68 on X and 507.32 on Y returns 1,024.78 whichever player wins, a profit of 24.78 on 1,000, before costs, limits, and execution risk.
How do you calculate arbitrage stakes?: Convert each best price to an implied probability with 1 / o and sum them to get S. For a total stake T, stake T x (1 / o) / S on each outcome. Every outcome then returns T / S, and the profit is T / S minus T. If S is 1 or above, there is no arbitrage.
Why does a three-way arbitrage need all three outcomes?: Because the guarantee only holds if every possible result is covered. In football match result markets the draw is an outcome, so leaving it out means the position loses everything staked whenever the match is drawn. Any market used for arbitrage must be mutually exclusive and exhaustive across the legs, including identical lines and periods.
Does exchange commission affect an arbitrage?: Yes, and it often removes it. Commission on net winnings lowers the usable price to 1 + (o - 1) x (1 - c). In the illustrative case on this page, a gap that looks like 0.74 percent on raw prices becomes a sum of 1.0054 after a 5 percent commission, which is no arbitrage at all.