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MasterMath

Arbitrage Betting Calculator

Check whether a set of odds lets you win whatever happens, how much to stake on each outcome, and what expected value a bet has against your own estimate.

Is there an arbitrage?

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Is there an arbitrage?—
Sum of the probabilities—
Bookmaker's margin—
Guaranteed profit—
Return—
Expected value of the first—
Your edge over the book—

How much to stake on each outcome

OutcomeOddsStakeYou would collect

How this was worked out

    Indicative result. Real arbitrages are rare, last minutes, and bookmakers limit or close accounts that work them systematically. This calculator is for understanding the arithmetic, not as a plan for making money.

    The formula

    an arbitrage exists if 1/c₁ + 1/c₂ + … < 1 · stake = total × (1/cᵢ) ÷ sum

    Where it comes from

    Every set of odds carries an implied probability, its inverse. If the ones across all outcomes add up to less than one, the market has left a gap: splitting the stake in proportion to those probabilities collects the same whoever wins, and that return beats what went in. When they add up to more than one, the excess is the bookmaker's margin.

    How to work it out by hand

    1. Work out the inverse of each set of odds and add them up
    2. If the sum is below 1, there is an arbitrage
    3. Split the total in proportion to each inverse
    4. Check that the return is the same in every case

    What is worth knowing

    The expected-value calculation is a different one and more useful in practice: it compares your own estimate of the probability against the one implied by the odds. If you think something happens 50 % of the time and you are paid 2.50, you have an edge even where no arbitrage exists. The problem is that estimating better than the bookmaker is far harder than it looks, and the margin works against you from the first unit staked.

    Frequently asked questions

    What is an arbitrage bet?

    A set of odds whose implied probabilities add up to less than 100 %, so that splitting the stake correctly wins whatever happens.

    How is the stake split?

    In proportion to the inverse of each set of odds: that way the return is identical whoever wins.

    Is sports arbitrage legal?

    Yes, but bookmakers forbid it in their terms and usually limit or close the accounts that practise it.

    What is the expected value of a bet?

    What you would win or lose on average per bet if you repeated it many times. It is positive only where your probability beats the one implied by the odds.