How to calculate an arb.

Start with decimal odds

The sums below use decimal odds: the total returned for every 1 staked, stake included. If your bookmaker shows something else, convert first.

Throughout this guide the example is a two-outcome match priced at 2.10 on one side by bookmaker A and 2.05 on the other by bookmaker B, with 1,000 to stake.

Is it an arb?

Divide 1 by each price. The result is the probability that price implies. Add them up and call the total S.

If S is below 1, the prices are an arb. If it is 1 or above they are not, and no way of splitting the stake will change that. The profit, as a share of everything you stake, is:

A common shortcut is 1 − S, which gives 3.60% here. It is close for small arbs and always a little low. The exact figure is 1 ÷ S − 1.

Split the stake

Each bet gets a share of the total in proportion to its implied probability:

Check it by multiplying back. 493.98 × 2.10 is 1,037.36 and 506.02 × 2.05 is 1,037.34: the same return whichever side wins, and 37 more than the 1,000 that went in. The shorter price takes the larger stake. That is the whole method:

Three outcomes

Nothing changes except the number of terms. A football match is priced 2.75 home at one bookmaker, 3.50 the draw at a second and 3.05 away at a third. You have 600.

About 614 back on 600 whatever the score. Open this example in the calculator.

Three bets mean three chances for a price to move before you finish, so a three-way arb needs a larger edge than a two-way one to be worth the trouble.

When one bet is already placed

In practice you rarely start from a total. You place the first bet, the one most likely to disappear, and then size the second to match what the first would return.

Both bets now return 1,050 on 1,012.20 staked. If the second price has moved by the time you get there, run the same sum with the new price before you bet. If the result is a loss, it tells you how large, and you can decide between taking it and letting the first bet stand on its own.

The price you need on the other side

Often you have one good price and are still looking for the other. It helps to know the number you are looking for: the price at which the pair exactly breaks even.

Anything above 1.909 on the other outcome makes an arb with 2.10, and anything below does not. Against 3.00 you need better than 1.50. Against 1.50 you need better than 3.00. The further above the break-even price you get, the bigger the arb.

With three outcomes, add up 1 ÷ odds for the two prices you hold, take that from 1, and divide 1 by what is left. Holding 2.75 on the home side and 3.05 on the away side, the two add up to 0.6915, which leaves 0.3085, so the draw has to be priced above 3.24.

Commission and other fees

A betting exchange takes a commission, typically a few percent, from the net winnings of a winning bet. Take it off the price before doing anything else:

Against 2.10 at a bookmaker, 2.08 on an exchange looks like a 4.50% arb. With 2% commission it is 3.95%. With 5% it is 3.12%. On thin arbs the commission is the difference between profit and loss, so never skip it. The calculator has a fee field for each bet.

Costs outside the bet belong in the sum as well: deposit and withdrawal charges, and currency conversion if a bookmaker holds your balance in another currency. A 2% arb paid for with a 2.5% conversion fee is a loss.

Rounding the stakes

Nobody bets 493.98 for fun, and bookmakers know it. Rounding each stake to a whole 5 or 10 makes the bets look ordinary, at a small cost: the two returns stop being equal.

The rule after rounding: multiply each stake by its odds, and make sure every result is still above the new total. On small totals a coarse step can eat the whole arb. With 70 to stake on prices of 1.30 and 4.60, a 1.4% arb, rounding to the nearest 10 gives 50 and 20, and the first outcome now returns 65 on 70 staked.

All the profit on one outcome

The even split is not the only one. You can size the other bets to return exactly your total, and put what is left on the outcome you favour.

If the 2.05 side wins you get your 1,000 back and nothing else. If the 2.10 side wins you get 1,075.60: the whole profit, about double the even split, on one result. See it in the calculator.

This is no longer a fixed profit. It is a bet you cannot lose on paper, which is a different thing. It is also useful when one bookmaker's limit is too low for the even split: the smaller covering stake may fit where the even one did not.

Before you trust the number

The arithmetic is never the part that fails. These are:

• Same event, same bet. The same match, the same market, the same line, and the same rules for a retirement, extra time or a postponement.
• Live prices. Read both on the bookmakers' own pages, not from memory or a screenshot.
• Limits. If one bookmaker will only take 60, size the whole arb from that bet, as in section 5.
• Too good. A result above roughly 10% is more often a pricing error than a gift, and bookmakers void bets struck at obvious errors.

The wider picture, including what bookmakers do about arbers, is in What is arbitrage betting? and the guide to bookmaker limits.