Key takeaways

• Every arb check comes down to one test: convert the best price on each outcome to implied probability and add them up. Under 100% is an arb.

• Implied probability is 1 divided by the decimal odds. For American odds, it's 100 / (odds + 100) for plus prices and |odds| / (|odds| + 100) for minus prices.

• The margin (100% minus the total) and the return on stake (1 / total minus 1) aren't the same number. A 98.02% total is a 1.98% margin and a 2.02% return.

• The same math strips the vig out of a single book's line. A -150/+130 market holds 3.48%, and the fair prices are -138 and +138.

The one test behind every arb

An arbitrage bet backs every outcome of an event, at whichever book has the best price on each one, with stakes sized so every outcome pays the same. Whether that's possible depends on a single number: the sum of the implied probabilities of the best prices.

If the sum is under 100%, the prices disagree enough that backing everything costs less than the payout. If it's 100% or more, no split of the stake guarantees a profit. Everything else in arbitrage betting, from stake sizing to choosing books, starts from this test.

Converting odds to implied probability

Implied probability is the break-even chance a price assumes. The formula you'll use depends on the format.

• Decimal odds: 1 / odds. A 2.10 price implies 1 / 2.10 = 47.62%.

• American plus odds: 100 / (odds + 100). +130 implies 100 / 230 = 43.48%.

• American minus odds: |odds| / (|odds| + 100). -120 implies 120 / 220 = 54.55%.

• Fractional odds: convert to decimal first by dividing and adding 1. 5/2 is 2.5 + 1 = 3.50, which implies 28.57%.

• Prediction market contracts: the price is the probability. A 47¢ contract implies 47%.

The odds converter handles all of these in both directions.

Worked example: decimal odds

One book prices Team A at 2.10 and another prices Team B at 1.95.

• Team A at 2.10: 47.62%

• Team B at 1.95: 51.28%

• Total: 98.90%

That's an arb. On a $1,000 total stake, each leg's stake is the total times its share of 98.90%:

• Team A: $1,000 × 47.62 / 98.90 = $481.48

• Team B: $1,000 × 51.28 / 98.90 = $518.52

Both outcomes return $1,011.11, so the locked profit is $11.11.

Worked example: American odds

Say Caesars has Team A at -120 and BetMGM has Team B at +130. These are illustrative prices.

• Team A at -120: 54.55%

• Team B at +130: 43.48%

• Total: 98.02%

On $500, the stakes are $278.23 and $221.77, and either outcome returns $510.08 for a $10.08 profit.

You'll get the same answer by converting to decimal first: -120 is 1.833 and +130 is 2.30, and 1 / 1.833 plus 1 / 2.30 is 98.02%.

Worked example: three-way market

Soccer moneylines have a draw, so there's a third term in the sum. Suppose the best prices across books are home 2.60, draw 3.50 and away 3.20.

• Home at 2.60: 38.46%

• Draw at 3.50: 28.57%

• Away at 3.20: 31.25%

• Total: 98.28%

On $1,000, the stakes come to $391.33, $290.71 and $317.96. Every outcome returns $1,017.47, a $17.47 profit.

Margin and return on stake

Arb tools quote two different percentages, and they're easy to mix up.

• The margin is 100% minus the total implied probability. In the American odds example, 100% − 98.02% = 1.98%.

• The return on stake is the guaranteed payout divided by the total stake, minus 1. That's 1 / 0.9802 − 1 = 2.02%, which matches the $10.08 profit on $500.

The return is always a little higher than the margin, and the gap grows as the margin grows. When you compare a scanner's number with your own calculation, check which one it's quoting. For sizing decisions, the return is the number that matches the money.

When the prices aren't an arb

A standard -110/-110 line at a single book implies 52.38% on each side, which adds up to 104.76%. Splitting $1,000 evenly returns $954.55 whichever side wins, a guaranteed $45.45 loss. That 4.76% is the book's hold.

Two books can disagree on a game and still sum above 100%, because each price carries its own book's hold. That's why a price with no vig built in, such as one from Novig, makes an arb easier to reach. Its side of the sum starts lower, so less disagreement is needed to get the total under 100%. The guide to arbing with prediction markets works through that with fees included.

Stripping the vig to find the fair price

The same math tells you what a book thinks the fair price is. Divide each side's implied probability by the total.

Take a -150/+130 line:

• -150 implies 60.00% and +130 implies 43.48%, a total of 103.48%. The hold is 3.48%.

• Fair probabilities: 60.00 / 103.48 = 57.98% and 43.48 / 103.48 = 42.02%.

• Fair prices: -138 and +138.

A price elsewhere that beats the fair price is a value bet, even when it isn't part of an arb. The no-vig panel in the odds converter runs this calculation for any two-sided market.

Arbitrage math questions

Why do arb percentages look so small?

Because the books' prices usually sit close to each other, and each one carries a margin. The examples in this guide return between 1.11% and 2.02% before fees.

Does the formula change with more than three outcomes?

No. Add one implied probability per outcome. The test and the stake split are the same for a two-way market, a three-way soccer line or a futures market with many outcomes.

Should I use decimal or American odds for the math?

Decimal is simpler, since implied probability is 1 / odds with no sign rules. Convert American prices to decimal first if you're doing it by hand.

How do fees fit in?

A per-contract fee on a prediction market raises the effective price. Add the fee to the price before converting it to implied probability. The stake sizing guide shows how much that changes the profit.

Check your own prices

Enter the best price on each outcome in the arbitrage calculator. It runs the 100% test, sizes the stakes and shows the profit.