acalculator

Is there an arbitrage in these odds?

Enter the best odds on each outcome and a total stake to see whether the odds leave an arbitrage and how to split the stake. The results are mathematics, not betting advice.

Your numbers

Odds format
Outcomes
Row 1
Row 2
Profit whichever outcome wins
$37.35

Spreading $1,000.00 across the outcomes pays $1,037.35 whichever wins, a profit of $37.35.

Return on the total stake
3.735%
Arbitrage?
Yes
Sum of implied probabilities
96.4%
Payout whichever outcome wins
$1,037.35
Stake on outcome 1
$493.98
Stake on outcome 2
$506.02

Profit whichever outcome wins: $37.35. Spreading $1,000.00 across the outcomes pays $1,037.35 whichever wins, a profit of $37.35.

The results are maths only. They are not betting advice or a promise of a return. Bet only what you can afford to lose. For help with gambling, call 1-800-GAMBLER. Terms of use

How to calculate

Checks whether the best odds on every outcome of one event add up to an arbitrage, and splits a total stake across the outcomes so each pays the same, with the profit or loss.

Example with the default inputs (Odds format American, Outcomes [Odds +110; Odds +105], Total stake $1,000.00): Spreading $1,000.00 across the outcomes pays $1,037.35 whichever wins, a profit of $37.35.

Method: With decimal odds dᵢ and total stake T: book B = Σ 1/dᵢ; stake on outcome i = T × (1/dᵢ) ÷ B; every outcome pays T ÷ B; profit = T ÷ B − T. An arbitrage exists when B < 1.

  • The outcomes listed cover every way the event can end, and exactly one of them wins.
  • Each bet is placed at the odds shown, with no limits, fees or cancelled bets. In practice odds move and bookmakers may void or limit such bets.
  • Stakes are exact; round them to what the bookmaker accepts, which changes the payouts slightly.
  • Results are mathematics, not betting advice.

Machine-readable copies: Markdown, JSON.

Worked examples

Each example is checked against the calculator on every build.

  1. Odds format American, Outcomes +110; +105, Total stake $1,000.00 gives Sum of implied probabilities 96.399535%, Arbitrage? Yes, Stake on outcome 1 $493.98, Stake on outcome 2 $506.02, Payout whichever outcome wins $1,037.35, Profit whichever outcome wins $37.35.Source: hand calculation in content.mdx: 10/21 + 20/41 = 830/861; stakes 1000 × (10/21) ÷ (830/861)
  2. Odds format Decimal, Outcomes 2.5; 3.6; 4, Total stake $100.00 gives Sum of implied probabilities 92.777778%, Arbitrage? Yes, Stake on outcome 1 $43.11, Payout whichever outcome wins $107.78.Source: hand calculation in content.mdx: 0.4 + 5/18 + 0.25 = 167/180; 100 × 180/167
  3. Odds format American, Outcomes -110; -110, Total stake $220.00 gives Arbitrage? No, Stake on outcome 1 $110.00, Payout whichever outcome wins $210.00, Profit whichever outcome wins -$10.00.Source: hand calculation in content.mdx: 22/21 > 1, so no arbitrage; 220 × 21/22 = 210

How it works

Each outcome’s odds become decimal odds dᵢ as exact fractions: American +A gives d = 1 + A ÷ 100, American −A gives d = 1 + 100 ÷ A (A ≥ 100), fractional a/b gives d = 1 + a ÷ b.

With total stake T:

  • Book B = 1/d₁ + 1/d₂ + … (exact). The sum of implied probabilities is 100 × B percent.
  • Stake on outcome i = T × (1/dᵢ) ÷ B.
  • Each outcome then pays stake × dᵢ = T ÷ B, the same for every outcome.
  • Profit = T ÷ B − T; return = 100 × (1 ÷ B − 1) percent.
  • It is an arbitrage (Yes) when the profit, computed as above, is above 0; in exact terms, when B < 1.

From 2 to 10 outcomes are allowed.

What you can type: spaces are ignored and − reads as -. Each format has its own shape, so odds typed in another format are never read as this one: the page says which format they are in and gives the same price in the chosen format. American odds have a + or − sign and a size of at least 100 (+150, -110, +1,000; commas are thousands commas). Decimal odds are a plain number above 1 with no sign (2.5); a single comma with no point is a decimal comma (2,5 is 2.5) unless exactly 3 digits follow it (1,000 is 1000), and any other comma is a thousands comma. Fractional odds are a/b, a:b, a-b or a to b with a and b above 0, or evens, even or evs (any letter case, to too); a bare number is not fractional odds. A number with no sign typed as American odds (100 or more), a whole number typed as fractional odds, or a number below 100 typed as a probability gets a message saying what to add (+150, 4/1, 40%). Decimal odds above 1,000,001 (American +100,000,000) give no answer.

Assumptions

  • The outcomes cover every way the event can end, and exactly one wins.
  • Every bet is placed at the odds shown, with no limits, fees, voids or rounding.

Worked examples by hand

+110 and +105, total 1,000. d₁ = 21/10 and d₂ = 41/20, so B = 10/21 + 20/41 = 830/861, or 96.40%: an arbitrage (Yes). Stake 1 = 1,000 × (10/21) ÷ (830/861) = 1,000 × 410/830 = 493.98; stake 2 = 1,000 × 420/830 = 506.02. Payout = 1,000 × 861/830 = 1,037.35, profit 37.35.

Decimal 2.5, 3.6 and 4, total 100. B = 0.4 + 5/18 + 0.25 = 167/180, or 92.78%. Stake 1 = 100 × 0.4 × 180/167 = 43.11. Payout = 100 × 180/167 = 107.78.

−110 and −110, total 220. B = 11/21 + 11/21 = 22/21, above 1: no arbitrage (No). Each stake is 110, each pays 220 × 21/22 = 210, a loss of 10.

A note on betting

These results are mathematics, not advice. Real bets carry risks the arithmetic leaves out: odds move, and bets can be voided or limited. If gambling is causing problems for you or someone you know, the US National Problem Gambling Helpline is 1-800-MY-RESET (call or text).

Other questions people ask

How do I know if odds are an arbitrage?

Add up 1 ÷ decimal odds over every outcome. If the sum is below 1 (below 100% in implied probability), the odds are an arbitrage. For +110 and +105 on the two sides of a game, the sum is 10/21 + 20/41 = 96.40%, so they are.

How do I split the stakes in an arbitrage?

Bet on each outcome in proportion to 1 ÷ its decimal odds. With a 1,000 total on +110 and +105, bet 1,000 × (10/21) ÷ (830/861) = 493.98 on the first and 506.02 on the second. Each pays 1,037.35.

How much profit does an arbitrage make?

The payout is the total stake ÷ the sum of 1 ÷ decimal odds, whichever outcome wins. The profit is the payout minus the total. At a 96.40% sum, the return is 1 ÷ 0.9640 − 1 = 3.73%.

What if the sum is over 100%?

Then there is no arbitrage. The same split still pays the same whichever outcome wins, but it loses money: at −110 on both sides (104.76%), a 220 total pays back 210.

Is arbitrage betting risk-free?

Only in the arithmetic. In practice odds can change before every bet is placed, a bookmaker can void or limit a bet, and rounded stakes change the payouts. Many bookmakers restrict accounts that bet this way.