We explain how to calculate arbitrage bets to give you an understanding of the maths behind arbitrage betting, and why it’s a lowrisk betting strategy. Learn how to calculate arbitrage bets between bookmakers and an exchange to maximise your potential arbitrage betting profit.
If you’re new to arbitrage betting we have already explained in detail what it is, but to provide context for this article, it is a trading technique which benefits from price differentials between two or more bookmakers or betting exchanges to guarantee a profit regardless of the outcome.
Arbitrage sports betting is the method of placing bets on all outcomes of an event at odds that guarantee a profit regardless of the eventual result.
Let’s explain this briefly. A fair market would be priced at 100% based on the likelihood of an event occurring, however, bookmakers will price their odds to go above a 100% probability, therefore giving them an edge.
Arbitrage opportunities are the reverse of this, whereby an arber will bet on all eventualities across a number of betting providers, giving them the opportunity to take advantage of discrepancies in price so the probability of the odds they have bet on is lower than 100%  therefore in their favour.
How to calculate arbitrage betting between a bookmaker and exchange
The most common arbitrage bet is made by taking positions in the market across a bookmaker and a betting exchange  backing at the bookmaker and then laying the same outcome on the betting exchange.
As an example let’s say you want to bet on a tennis match between Player A and Player B. The odds on the bookmaker and the market implied probabilities in brackets  learn how to calculate betting margins  are displayed below:
Player A to win 
Player B to win 
Market margin 

Odds 
2.20 (45.45%) 
1.68 (59.52%) 
105% 
You now look at the lay price on Player A to win  betting that he won’t win  which is 1.98 with the Smarkets exchange. The table below highlights this is an arbitrage opportunity as the combined implied probability for bets is 95.96%.
Player A to win at bookmaker 
Player A not to win at Smarkets 
Combined market margin 

Odds 
2.20 (45.45%) 
1.98 (50.51%) 
95.96% 
Calculate your lay stake for arbitrage bets on a betting exchange
Now you stake £200 on Player A to win on the bookmaker with odds of 2.2. To calculate your lay stake on the Smarkets exchange you simply use the following calculation:
(back price x back stake) / (current lay odds  exchange commission)
You would then make a £224.49 lay bet on Player A to win at odds of 1.98 on the betting exchange.
Calculate your profit if you win at the bookmaker
Calculating your profit if your bet wins with the bookmaker is simple:
Profit = (back odds  1) * back stake  (lay odds 1) * lay stake
Calculate profit if the lay bet wins at the exchange:
Next, you calculate your profit if your lay bet with the exchange is successful:
Profit = (lay stake * (1  commission))  bookmaker stake
Overall arbitrage profit
Your overall arbitrage profit regardless of the outcome is displayed below :
Outcome 
Bookmaker 
Smarkets exchange 
Profit/loss 
Player A wins 
+£240 
£220 
+£20 
Player A loses 
£200 
+ £220 (includes 2% commission) 
+£20 
So if you placed this arbitrage bet, you would guarantee a profit of £20 irrespective of Player A winning or not.
How to calculate arbitrage between two bookmakers
Before betting exchanges arbitrage bets were made between two or more bookmakers  these are becoming less prevalent because most bookmakers have very similar odds. Having said that they do still occur.
For an arb to occur between bookmakers you need to identify the two bookmakers offering the highest odds on each outcome. The table below can help you quickly recognise if there is a potential arb across bookmakers. If you find odds similar to outcome 1, then odds which are greater than what is identified in outcome 2 would present an arbitrage opportunity:
Outcome 1 odds 
Outcome 2 odds 
1.2 
>6 
1.3 
>4.33 
1.4 
>3.5 
1.5 
>3 
1.6 
> 2.66 
1.7 
> 2.42 
1.8 
> 2.25 
1.9 
>2.11 
2.0 
> 2.0 
As an example let’s say you want to bet on a darts match between Player A and Player B. The odds on both bookmakers and the market implied probabilities are displayed below:
Player A to win 
Player B to win 
Market margin 

Bookmaker A odds 
1.30 (76.92%) 
3.70 (27%) 
104% 
Bookmaker B odds 
1.40 (71.43%) 
2.98 (33.56%) 
105% 
This clearly shows an arbitrage opportunity between bookmaker A and bookmaker B. The table below combines the odds and implied probabilities and shows they are under 100%  or in other words, in your favour  learn how to convert odds in to implied probability.
Player A to win at bookmaker B 
Player B to win at bookmaker A 
Combined market margin 

Odds 
1.40 (71.43%) 
3.70 (27%) 
98.43% 
By crossmatching the odds for each outcome across the two bookmakers the combined market margin is now in the bettor's favour providing a guaranteed return of 1.57%.
Calculating how much to stake for arbitrage bets
Now you must calculate how much to stake with each bookmaker to guarantee an equal return without consideration for the result. Let’s say you are willing to stake £100 overall.
Stake at each bookmaker = (Overall stake * Bookmaker implied probability)/Combined market margin
Overall arbitrage profit
Your overall arbitrage profit regardless of the outcome is displayed below :
Bookmaker A 
Bookmaker B 
Profit/loss 

Player A wins 
£27.43 
+£29.02 
+£1.59 
Player B wins 
+£74.06 
£72.57 
+£1.46 
How to calculate arbitrage on threeway betting markets
Now you know how to calculate arbitrage opportunities for a simple twoway arbitrage bet, you can look at threeway or 1X2 arbs. Threeway arbs are more prevalent in football where there is a possibility of a draw.
When arbing with a betting exchange there is no need for threeway arbs as you could back Manchester United to win on the bookmaker against Liverpool and then lay them to win on the exchange  which covers anything other than a Liverpool win.
So this example is for a threeway arb between three separate bookmakers. Let’s say you find the following odds on a game between Manchester United and Liverpool:
Man United to win 
Draw 
Liverpool to win 
Market margin 

Odds 
Bookmaker A 2.12 (47.17%) 
Bookmaker B 3.55 (28.17%) 
Bookmaker C 5.40 (18.52%) 
93.86% 
This highlights an arbitrage opportunity between the three bookmakers as the table above combines the odds and implied probabilities and shows the market margin is under 100%.
Calculating how much to stake for arbitrage bets
Now you need to calculate how much to stake with each bookmaker to guarantee an equal return depending on which bookmaker you win with. Let’s say you want to stake £1500 overall.
Stake at each bookmaker = (Overall stake * Bookmaker implied probability)/Combined market margin
Overall arbitrage profit
Your overall arbitrage profit regardless of the outcome is displayed below :
Outcome 
Bookmaker A 
Bookmaker B 
Bookmaker C 
Profit/loss 
Man United win 
+£844.30 
£450.19 
£295.97 
£98.14 
Draw 
£753.84 
+1147.98 
£295.97 
£98.17 
Liverpool win 
£753.84 
£450.19 
+1302.27 
£98.24 
So for this example, if you placed £1500 stake across the different bookmakers as suggested you would guarantee a profit of around £98.
Apply this to betting
Now you know how to calculate arbitrage bets you are in the best position to take advantage of price discrepancies across bookmakers and betting exchanges. Remember Smarkets accept arbitrage bettors and also offer the best odds online  with our industrylow 2% commission.