How to Calculate Arbitrage Bets: Formula, Stakes, Examples
The surebet formula and stakes for 2-way and 3-way bets: from a bankroll, a leg or a limit, with the Polymarket fee, a free bet, rounding and a hedge.
Contents
An arbitrage bet is calculated in two steps: add up the inverse odds — if the sum is below one, there is a surebet — and split the bankroll in proportion to the inverse odds, so the payout is the same whatever the outcome. By hand you use the formulas below; the arbitrage calculator does the same for 2 to 10 outcomes, accounts for a leg's commission and recalculates the stakes from a leg you have already placed.
In arbitrage betting you calculate before the first bet, not after it: in Dozor data for October 6 – November 5, 2026, half of 4,829,879 surebet episodes lasted 2 seconds or less, and 15.9% lasted over 10 seconds. The median ROI was 3.2%, and 8.4% of episodes had an ROI of 10% or more — why those are worth a second check is at the end.
Formula and check
For outcomes with odds k₁, k₂ (and k₃ for three outcomes):
Σ = 1/k₁ + 1/k₂ (+ 1/k₃)
If Σ < 1, there is a surebet. ROI equals 1/Σ − 1.
| Odds | Σ | Surebet | ROI |
|---|---|---|---|
| 2.10 and 2.05 | 0.964 | yes | 3.73% |
| 1.95 and 2.00 | 1.013 | no | −1.27% |
| 2.10, 3.60 and 4.20 | 0.992 | yes | 0.80% |
The second row is a negative surebet: betting both outcomes locks in a small loss, used only to clear a bonus.
Check your own odds in the arbitrage calculator: the Σ 1/k row shows whether there is a surebet.
Stakes for two outcomes
The stake on leg i equals bankroll × (1/kᵢ) / Σ. That makes the payout of every leg the same.
Bankroll $10,000, odds 2.10 and 2.05, Σ = 0.963995:
| Leg | Odds | Stake | Payout |
|---|---|---|---|
| 1 | 2.10 | $4,939.76 | $10,373.49 |
| 2 | 2.05 | $5,060.24 | $10,373.49 |
The profit is $373.49 on any outcome, an ROI of 3.73%. The margin of safety is small: it stays a surebet while the first odds are at least 1.952 with the second at 2.05, or the second at least 1.909 with the first at 2.10. The threshold is 1 / (1 − Σ of the other 1/k).
Three outcomes
Same formula, three terms. Odds 2.10, 3.60 and 4.20, bankroll $1,000, Σ = 0.992063:
| Outcome | Odds | Stake | Payout |
|---|---|---|---|
| 1 | 2.10 | $480.00 | $1,008.00 |
| 2 | 3.60 | $280.00 | $1,008.00 |
| 3 | 4.20 | $240.00 | $1,008.00 |
The profit is $8, an ROI of 0.80%. Three legs at three books have to be placed before any of the three prices moves. When all outcomes come from one book and you want an equal payout, that is dutching — there is a dutching calculator for it.
From one leg or from a limit
From one leg. If you place a fixed amount F at one book, the other stakes equal F × kᵢ / kⱼ. Leg 1 is $5,000 at 2.10, so leg 2 = 5,000 × 2.10 / 2.05 = $5,121.95. Total $10,121.95, payout $10,500, profit $378.05. In the calculator, this is the "Stake on one leg" mode.
From limits. Book 1 accepts up to $3,000, book 2 up to $5,000. The leading leg is the one where limit × odds is smaller: 3,000 × 2.10 = 6,300 versus 5,000 × 2.05 = 10,250, so the first. The stakes are $3,000 and $3,073.17, bankroll $6,073.17, profit $226.83: you cannot place more without exceeding a limit.
In the Dozor scanner, the amount for each book is set in the "Stake at this bookmaker" field: the calculator in the surebet card fills it in and works out the other stakes from it.
With an exchange fee
Polymarket charges its taker fee on top of the share price: on sports markets it is shares × 0.05 × price × (1 − price). So a share's net odds are lower than 1 / price: k′ = 1 / (price × (1 + 0.05 × (1 − price))).
Example: a sportsbook offers 2.20, the opposite outcome on Polymarket costs 52¢.
| Without the fee | With the fee | |
|---|---|---|
| Share odds | 1.923 | 1.878 |
| Σ | 0.9745 | 0.9870 |
| ROI | 2.61% | 1.31% |
The fee took half the return. From a $1,000 budget, the stakes are $460.52 at the sportsbook and 1,013.14 shares for $539.48, of which $12.64 is the fee; the profit is $13.14 on any outcome. The surebet holds while a share costs no more than 53.30¢.
In the arbitrage calculator, set the leg type to "Exchange ¢", enter the 52¢ price and the venue — Polymarket with the market category: the calculator adds the fee itself and uses k′ = 1.878. The Polymarket arbitrage calculator starts with an exchange leg by default. Kalshi has its own fee formula; it is covered in the article on Polymarket and Kalshi arbitrage with sportsbooks.
With a free bet
A free bet is calculated differently from a regular bet. If the promotion's terms do not return the free bet stake with the winnings, its payout is stake × (k − 1). The bets on the other outcomes are worked out from that payout: sⱼ = stake × (k − 1) / kⱼ. The profit is stake × (k − 1) − Σ sⱼ. If the stake is returned, the free bet is calculated as a regular bet paying stake × k. Which applies to your free bet is stated in the promotion's terms.
Example: a $1,000 free bet at 5.00 against 1.25 pays $4,000, the other outcome takes $3,200, and the profit is $800 whatever the outcome, a free bet conversion of 80%. In the arbitrage calculator, set the leg type to "Free bet" and enter its amount; it works out the other stakes.
When the second leg moves
The first leg is placed: $4,940 at 2.10, payout $10,374. While you were opening the second book, its odds dropped from 2.05 to 1.85.
| What to do | Stake on the second outcome | Result on any outcome |
|---|---|---|
| Hedge at 1.85 | $5,607.57 | −$173.57 |
| Find 1.97 at another book | $5,265.99 | +$168.01 |
The threshold is 1.909: above it a hedge still gives a profit, below it a fixed loss. Not hedging means keeping a regular bet: if it loses, you lose the whole $4,940. The hedge stake is first leg payout / new odds; the hedge calculator does it for you; with three or more outcomes, use the "Already placed" tab of the arbitrage calculator. To get into this less often, place first the leg whose odds move more often or whose limit is lower, and watch the price age in the surebet card.
Rounding
Stakes like $4,939.76 and $5,060.24 look calculated, so people round them. Bankroll $10,000, odds 2.10 and 2.05, step $100:
| Stakes | Total | Profit, outcome 1 | Profit, outcome 2 |
|---|---|---|---|
| 4,900 and 5,100 — to the nearest hundred | $10,000 | $290 | $455 |
| 4,900 and 5,000 — trying down and up | $9,900 | $390 | $350 |
| 5,000 and 5,100 — one step over the bankroll | $10,100 | $400 | $355 |
Rounding "to the nearest" gives a worst case of $290; trying each stake down and up within the bankroll gives $350, an actual ROI of 3.54%. The rounding step is set in the calculator settings.
ROI and margin
The margin is one book's markup: Σ − 1 over its own odds. At 1.90 on both outcomes the margin is 5.26%, at 2.10, 3.40 and 3.60 on three outcomes it is 4.81%; sometimes it is calculated as 1 − 1/Σ, which gives 5.0% for 1.90 and 1.90. A surebet's ROI is 1/Σ − 1 over the best odds at different books: the same sum of inverse odds, but below one. You can work out your book's margin in the margin calculator.
Contract price and odds
On prediction markets a share's price is a probability in cents, and the odds equal 1 / price: 52¢ is 1.923, 46¢ is 2.174. You can convert a price into odds and back in the odds converter; it does not include fees, which the formula in the section above adds.
Why an ROI above 10% is a warning sign
| Surebet ROI | Episodes | Share | Over 60 s |
|---|---|---|---|
| below 1% | 561,944 | 11.6% | 6.6% |
| 1–2% | 975,535 | 20.2% | 4.7% |
| 2–3% | 746,510 | 15.5% | 4.0% |
| 3–5% | 997,481 | 20.7% | 3.6% |
| 5–10% | 1,142,541 | 23.7% | 2.5% |
| 10% and above | 405,868 | 8.4% | 2.6% |
What it means: the higher the ROI, the faster the surebet disappears — 6.6% of surebets below 1% hold for over a minute versus 2.6% of those at 10% and above. A large gap is a reason to check the price age and both books' market rules: a book may void a bet placed at an obvious error in the odds (Pinnacle's general rules say so), and then the second leg is left as a regular bet. The scanner does not show surebets above 15% at all: a gap that large is almost always built from stale prices.
FAQ
Do I have to bet the exact calculated amounts?
No. Rounding to hundreds changes the profit by tens of dollars; it is better to try rounding each stake down and up and take the option with the best worst case, as in the table above.
How do I calculate if a limit is below the calculated stake?
From the limit: the leading leg is the one with the smaller limit × odds, and the other stakes equal limit × k₁ / kⱼ.
What if the second leg did not go through?
Calculate a hedge: the stake on the second outcome = first leg payout / current odds. If the odds are above the threshold k₁ / (k₁ − 1), the hedge gives a profit; below it, a fixed loss instead of the risk of losing the whole first bet.
How do I calculate a surebet with a free bet?
The payout of a free bet without the stake returned is stake × (k − 1); the bets on the other outcomes are that payout divided by their odds.
How is ROI different from margin?
The margin is one book's markup over fair probabilities, a surebet's ROI is the return on betting the best odds at different books. More in the glossary.
Sources
Checked on November 5, 2026.
- Polymarket fees: the taker fee is shares × feeRate × price × (1 − price), feeRate on sports markets is 0.05.
- Pinnacle betting rules: bets placed at an obvious error in the odds may be voided.
- Dozor: data — methodology.
What next
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