Arbitrage betting: the maths and why margins block it

Arbitrage betting spreads stakes over every outcome so the money returned is the same whichever outcome occurs. It exists only when the implied probabilities (1 divided by each decimal price), taken at the best prices across different bookmakers, add to less than 1. One bookmaker's own prices add to more than 1, because of margin.

By BetSmart Kenya Editorial (prepared with AI tools under our editorial policy) · Last updated 2026-10-26

The short answer

Arbitrage is a set of bets on every possible outcome of one event, with stakes sized so that the amount returned is identical whichever outcome occurs. The test uses implied probabilities, as explained in the guide on converting betting odds to probability. If the best prices across different bookmakers give implied probabilities that add to less than 1, the set returns more than it costs on paper.

A single bookmaker's prices for one market add to more than 1. That excess is its margin, so one book on its own cannot be turned into an arbitrage set. This guide shows the derivation with numbers you can check, then lists the frictions that stand between a sum below 1 and an actual result. It describes the arithmetic only and does not suggest anyone should do it.

The derivation with decimal odds

Take an event with n outcomes. Outcome i has decimal price D_i, and a stake of s_i on it returns s_i × D_i if it occurs. We want the same return R whichever outcome occurs.

Here S is the sum of the implied probabilities, each being 1 ÷ D_i. The gain is positive only if S is below 1. If S equals 1 the gain is zero, and if S is above 1 the same-result set is a certain loss of T × (S − 1) ÷ S.

Why one bookmaker's prices add to more than 1

A bookmaker sets prices so that its implied probabilities add to more than 1. The amount above 1 is the margin, covered in the guide on what overround is. With a margin, staking on every outcome at that one bookmaker returns less than the total staked, as the derivation shows for S above 1.

Illustrative example. A two-outcome event has outcomes X and Y. Bookmaker A offers X at 2.50 and Y at 1.60. Bookmaker B offers X at 1.60 and Y at 2.00. The prices are invented, and they are chosen so the arithmetic is exact.

Both sums are above 1, so neither bookmaker alone gives a same-result set with a gain. The method for the margin figure is in the guide on calculating the bookmaker's margin.

The condition across books

The bookmakers in the example disagree about X and Y. Taking the best price for each outcome, from whichever book offers it, gives four combinations.

Only the third row has S below 1. That row is the whole condition: the best price on each outcome must come from different books, and their implied probabilities must add to less than 1. The more tightly the books agree, the closer S sits to the margin of a single book, and the less often the sum drops below 1.

Sizing the stakes

Illustrative example. Use the third row with a total stake T of KES 900. From the formula, s_i = T × (1 ÷ D_i) ÷ S.

The two stakes add to KES 900. Either outcome returns KES 1,000, so the gain is KES 100, which is 100 ÷ 900 = 11.1% of the total stake. The formula gives the same: 900 × (1 − 0.9) ÷ 0.9 = 100. A gap this wide is chosen for easy arithmetic; the example says nothing about how big or how common such gaps are, and none was measured for this guide.

Frictions, stated as rules

The table above assumes every bet is placed at the shown price and settles as expected. In practice each assumption can fail. The rows below rework the example to show the effect of each failure, with exact numbers.

Three rules follow. First, a same-result set exists only while all its legs are in place, so until the last one is accepted the person holds an ordinary bet. Second, a gain scales with the smallest leg the limits allow: see the guide on maximum stake and payout limits. Third, whether a void leg returns the stake and what a cancelled or corrected bet means are set by the operator's published rules, as the guide on void bets and pushes explains.

Rounding is a smaller but real friction. Stakes in the example are whole shillings, which real stakes need not be, and a stake rounded away from its formula value breaks the equal-return property slightly. How payouts themselves are rounded is covered in the guide on how betting payouts are rounded in Kenya.

What Kenyan law says about it

No provision on arbitrage was found in the Gambling Control Act, 2025 or the Conduct of Gambling Operations Regulations, 2026 read for this guide. Three general rules still shape the arithmetic. Section 71 of the Act says a player in an online gambling activity shall not bet less than twenty shillings, so every leg must be at least KES 20 (see the guide on the minimum online stake).

Regulation 23(1) requires a licensee to clearly display betting odds, rules and terms applicable to each bet, and r.23(2) says it may not alter the terms of a bet after it has been accepted, except as permitted by law or approved rules. Regulation 24(1) requires winning bets to be settled promptly and in accordance with the published rules. These texts do not decide whether a particular bet is voided or corrected, so each operator's rules matter. Check the live GRA register for who is licensed.

Safer-play note

1. Treat any apparent gap as a calculation to understand, not as a plan. Until every leg is placed and settled, there is a risk of loss, as the table of frictions shows.
2. Using several operators means money held and staked in several places, which can raise total spending beyond what you meant to commit. Keep the total inside the budget you set in advance, as in the guide on setting a betting budget and limits.
3. Do not borrow or use money meant for rent, school fees or bills to fund stakes, and do not respond to a loss by increasing stakes to recover it.
4. If betting is causing harm, the GRA toll-free line and the self-exclusion process are on the responsible gambling page.

Betting is not a way to make money: every price carries a cost, and an exact calculation on paper is not a result. 18+. Gambling is addictive. Play responsibly.