Is arbitrage betting profitable? A math-first answer

A complete set of accepted prices can have a positive theoretical return when reciprocal decimal odds sum below one; fees, different settlement terms or failed execution can erase that edge.

The short answer

Is arbitrage betting profitable? It can have a positive theoretical return when prices cover every possible outcome and the sum of their reciprocal decimal odds is below one. That result assumes the quoted prices are accepted, the bets settle under compatible rules, and relevant costs are included. A calculator cannot guarantee that those conditions will hold in practice.

The first useful question is therefore not “How much can I make?” It is “What would have to be true for this complete position to return more than its total cost?” The equations below answer the mathematical part. They do not predict how often a suitable opportunity will appear or whether a bookmaker will accept a stake.

The condition behind a theoretical arbitrage

For decimal odds (O_1, O_2, \ldots, O_n), calculate:

S = 1/O₁ + 1/O₂ + ... + 1/Oₙ

The pre-cost condition is S < 1. If the outcomes are mutually exclusive and collectively cover every possible settlement, a total budget B can be split as:

Stake on outcome i = B × (1/Oᵢ) / S
Equal gross payout = B / S
Theoretical profit = B × (1/S − 1)
ROI on total stakes = (1/S − 1) × 100%

These equations describe a hypothetical allocation. They do not account automatically for exchange commission, deposit or withdrawal costs, currency conversion, stake increments, rejected bets, or settlement differences.

A two-outcome illustration

Suppose two hypothetical decimal prices are both 2.10 and you choose a total budget of 200 units. The reciprocal sum is about 0.952381. Equal stakes of 100 units create a gross payout of 210 units whichever side wins, so the pre-cost theoretical profit is 10 units, or 5% of the 200-unit stake total.

The probability gap is 1 − S, about 4.762%. That is not the same as ROI on stakes, which is 1/S − 1, about 5%. A software display may use either convention. Check the definition before comparing percentages.

A three-outcome illustration

A 90-minute football 1X2 market has home, draw and away outcomes. If the hypothetical prices are 2.20, 3.80 and 4.10, their reciprocal sum is about 0.961605. For a 1,000-unit budget, the approximate allocations are:

The small payout differences come from rounding the displayed stakes. The mathematical result is only useful if the three prices refer to the same event, market period and settlement terms, and each stake is accepted at the stated amount.

Why a positive calculation can still lose money

The formula has a narrow job: it tests prices for a complete set of outcomes. It does not verify the inputs or execute the bets. Check these failure points before treating the displayed edge as meaningful:

• A price changes before acceptance. Recalculate using the price on the accepted ticket, not the price on an earlier screen.
• The outcomes are incomplete. Two sides of a three-way match result leave the draw uncovered. A set of selected correct scores does not cover every score.
• The markets are not equivalent. A 90-minute result, qualification market, handicap and total can settle differently even when they refer to the same match.
• A stake is rounded or limited. Recompute the net result for every outcome using the amounts the operator actually accepts.
• Costs are missing. Include applicable commission and funding or currency costs. Compare net accepted positions, not a gross percentage.
• One leg fails. If only one ticket is accepted, the planned hedge is not complete. You are left with a position whose risk must be assessed on its own.

Smarkets' calculation guide explains the reciprocal-probability method and provides an exchange example. Its page is a mathematical reference, not evidence that any example price remains available now.

How to evaluate an opportunity

1. Record the event, participants, start time, market period and settlement rules.
2. Confirm that the selections cover every possible outcome without overlap or gaps.
3. Check each price directly at the relevant operator and note the time.
4. Calculate S, then allocate stakes using the same total-budget convention.
5. Apply fees, commission, currency conversion and feasible stake increments.
6. Work out the net result for each settlement outcome after rounding.
7. Treat only confirmed tickets as accepted positions and save the details.

If the lowest net outcome is not positive after realistic costs and accepted amounts, the original theoretical edge is not a practical profit estimate. You can reject the position rather than chase a changed price.

Include costs in the outcome, not only in the quote

For a simple illustration, let C be the total costs you attribute to a completed position. If all stakes are accepted and costs do not depend on which result occurs, the net amount is:

Theoretical gross profit = B × (1/S − 1)
Net profit after fixed costs = B × (1/S − 1) − C

Suppose hypothetical odds produce S = 0.98 and the total stake budget is 1,000 units. The equalized gross payout is about 1,020.41, so the theoretical profit is about 20.41 units before costs. If 25 units of relevant charges are attributable to that position, the net is about −4.59 units. The prices still satisfy the pre-cost condition, but this example is not profitable after the stated costs.

Some costs depend on the outcome or the amount won. In that case, calculate each result separately rather than subtracting one average fee. For outcome i, start with its accepted gross payout, subtract all accepted stakes, then apply the charges that actually apply to that settlement. Include currency conversion, funding friction or a share of software cost only when you can measure and reasonably allocate it. Do not count an entire monthly subscription against one position and then ignore it in every other record.

Re-check the position when a price moves

The allocation is tied to the odds used in the calculation. A small change can remove the margin or create a negative outcome. Consider two hypothetical decimal prices that were both 2.10. With a total budget of 200 units, the initial allocation is 100 units on each side, producing a 210-unit gross payout whichever side wins.

If the second price falls to 1.90 before its ticket is accepted, keeping the 100-unit stake gives a 190-unit payout on that side. The first outcome returns 210 units, but the second returns only 190 against 200 units staked. The position now ranges from a 10-unit gross profit to a 10-unit gross loss before any costs. Recalculate with the new quote and the amount the operator will actually accept; do not keep using the earlier percentage.

This is why a useful record includes the original alert time, the price when each ticket was accepted, the accepted amount, and the resulting payout by outcome. An alert screenshot alone does not establish a completed hedge. If the second leg is unavailable, treat the first accepted bet as its own exposure; do not describe the intended two-leg calculation as the result.

What this page can and cannot rank

The saved source material supports the calculation method, but it does not provide a representative dataset of sports, average margins, account limits, or opportunity frequency. It cannot support a claim that football, tennis, basketball or another sport is consistently “best” for arbitrage. Compare complete, equivalent markets and your own dated observations instead of assuming a sport-wide advantage.

Likewise, no formula establishes that a scanner will save enough money to justify a subscription. Use the software comparison to assess provider claims and the calculator guide for worked stake allocations. A single-selection positive-EV estimate is a different question; see the positive-EV tool guide. Legal status depends on jurisdiction and the specific service; see the dedicated arbitrage legality guide rather than treating a positive calculation as legal advice.