April 21, 2020

Arbitrage Betting Formulas & Calculations | Worked Examples

Upon learning about arbitrage betting, it is easy to assume that the formulas needed to calculate the necessary stakes for each bet must be quite complicated!

​However, the formulas and calculations are in fact very simple.

Below, I will outline these simple, yet highly effective formulas ​so that you can determin​e whether an arbitrage opportunity exists and if so, how to calculate how much you should stake on each side to ensure a guaranteed profit!

Soon enough, arbitrage calculations will be second nature to you!

Theory behind arbitrage betting ​formulas

How to determine if an arbitrage opportunity exists

Before we dive into the examples, it is important to understand what the odds that the bookmakers provide actually represent.

In essence, these odds are an expression of probability; that is, they represent the likelihood of an outcome occurring (in the eyes of that bookmaker). In theory, odds of 2.00 in a 2 way bet (Win or Loss being the only outcomes), represent a 50% chance of that outcome occurring.

If you stake $100 and win, you get $200, otherwise you get $0 (net profit is either +$100 or -$100).

The way to calculate the probability based on the odds is to calculate the inverse of the decimal odds. In this case:

Arbitrage Calculation

A probability of 0 represents 0% chance and a probability of 1 represents 100% (a sure outcome). So ½ represents a 50% chance.

Now, we do not live in a theoretical world and in reality the bookmakers are trying to make money, so even if they think that there is a 50% chance of an outcome occurring, they will not set odds of 2.00. They will set them at 1.90 or similar to build in a small profit margin.

This means that when calculating the percentage chance of each outcome and summing them up, the total probability will be greater than 100%.

Let’s take a look at a line betting example:

  • Bookmaker A: Broncos (-6.5) @ 2.00, Cowboys (+6.5) @ 2.00

The two possible outcomes from this bet are:

  • Broncos win by 7 or more points.
  • Broncos win by 6 or less points, draw or lose.

If we sum up the probabilities of each option, we get:

Arbitrage Calculation

As expected, in theory the probabilities for each option should sum to 100%. If we use a more realistic example, where there is a bookmaker profit margin built in, we will see that the total percentage exceeds 100%.

  • William Hill: Broncos (-6.5) @ 1.90, Cowboys (+6.5) @ 1.90

If we sum up the probabilities of each option, we get:

Arbitrage Calculation

As an arbitrager, this is an undesirable situation as no matter how much we stake on each side, there is no way to guarantee a profit regardless of the outcome.

So in essence, arbitragers are searching for odds that result in probabilities which sum to less than 100%.

“An arbitrage opportunity exists when the implied probabilities (from the odds) sum to less than 100%.”

Furthermore, the lower the percentage, the greater we can profit from that arb.

How to calculate how much to stake on an arb:

This part is even simpler than determining whether an arb exists. Simply choose your desired total winnings (across all bets), then divide the total winnings by the odds for each bet.

Arbitrage Formula

Where A and B are the two sides of the bet. This gives you the necessary stake for to ensure a guaranteed profit.

Let’s have a look with an example.

Example: 2 way arbitrage betting

2 way arbitrage betting refers to betting on a match with only 2 outcomes (typically Win/Loss or Over/Under as per the earlier example).

Let’s look at a theoretical tennis match between Dan Evans and Nick Kyrgios. 12Bet and Sportsbet are both offering head-to-head (outright win) markets for the match.


Dan Evans

Nick Kyrgios

Probability

12Bet

1.80

2.15

102.1%

Sportsbet

1.68

2.30

103.0%

Arbitrage

1.80 (12Bet)

2.30 (Sportsbet)

99.0%

From the example above, it is clear that by selectively taking the best odds from the two bookmakers, we can form an arbitrage opportunity.

Arbitrage Tennis

The 99% is important, as it indicates how much we can profit from the arb. 99% means that for every 100 units of winnings we receive, 99 units are required as a stake.

Let’s perform the stake calculation (using the formula we discussed earlier) for this arbitrage opportunity to see this in action.

We will assume total winnings of $500.


Odds

Stake

Potential Winnings

Dan Evans

1.80

$277.78

$500

Nick Kyrgios

2.30

$217.39

$500

Total


$495.17


Hopefully you can see that the stake of $277.78 for Dan Evans was calculated by:

Arbitrage Formula Stake

And similarly for Nick Kyrgios:

Arbitrage Formula

“Your stake is always inversely proportional to the odds for that leg. Higher odds mean lower stakes.”

As an aside, it is a terrible idea to bet such a precise amount like $277.78 as it screams arbitrage betting to the bookmaker. No regular punter bets like this.

Anyway, in the example above, we are staking $495.17 to win $500, which is a net profit of $4.83. Not an amazing profit, but still a genuine arb.

Interested in taking a test to see how well you can spot a 2-way arbitrage opportunity? Click ‘start quiz’ below!

The table below is a ‘cheat sheet’ to help give you a feel for the odds required for an arbitrage opportunity to arise in 2 way betting.

As you can see, as the odds for the favourite approach 1, the odds for the underdog must be ever larger for an arb to exist.

Favourite Odds

Underdog Odds

1.10

11.00

1.15

7.67

1.20

6.00

1.25

5.00

1.333

4.00

1.50

3.00

1.6660

2.50

1.75

2.33

2.00

2.00

Example: 3 way arbitrage betting

As you may have guessed by now, 3 way betting occurs when there are 3 possible outcomes for a betting market.

In soccer, a team can win, lose or draw (3 outcomes). Despite the extra outcome compared to 2 way betting, the process is fundamentally the same. Sum the probabilities of each of the outcomes and see if they are less than 100%.

Let’s look at an example. Manchester United vs. Arsenal in the Premier League.


Man U

Draw

Arsenal

Probability

William Hill

1.60

3.00

8.00

108.3%

Nordic Bet

1.80

3.20

5.00

106.8%

SBO

1.90

3.10

4.00

109.9%

Arbitrage

1.90

3.20

8.00

96.4%

By taking the best possible odds from each of the bookmakers, we are able to find an arbitrage opportunity.

Arbitrage Betting Formula

Note that even if Nordic Bet had odds of 1.90 for Manchester United to win, you should not bet with them for this outcome as you are already betting on the draw with them.

Betting on Man U as well would flag your account in their systems as a potential arbitrage trader.

Now, let’s perform the stakes calculation based on total winnings of $1,000 in this arbitrage bet.


Odds

Stake

Potential Winnings

Man U

1.90

$526.32

$1,000

Draw

3.20

$312.50

$1,000

Arsenal

8.00

$125

$1,000

Total


$963.82


Again, we are simply dividing the total winnings ($1,000) by the odds for each outcome to determine the stake for that leg of the bet.

As we mentioned earlier, a bet of $125 on Arsenal is fine, but betting $526.32 on Manchester United or $312.50 is not advised as it is the kind of bet only someone practising sports arbitrage would do.

Football

Anyway, in this example, we are wagering $963.82 for guaranteed winnings of $1,000. This is a tidy net profit of $36.18. Not bad for a few minutes of work!

The major takeaway from this example is that it is absolutely fundamental that you bet on the draw as well as each of the sides. If you forget to do so, you have the potential for big losses, which can wipe out your hard work on many arbs.

Forgetting to bet on the draw is a common mistake for new arbers, who are so excited thinking that they have found a huge arb that they forget to cover all outcomes of the match!

Now, let’s test your knowledge! Click ‘start quiz’ below to see if you can spot arbitrage opportunities in 3 way betting!

Free online arbitrage calculators

Now that you understand the basics of arbitrage calculations, let me introduce you to a few free online arbitrage calculators.

Rather than doing all of the calculating manually yourself, it is possible to select your bet type (2 way, 3 way etc.) and enter the best possible odds that you can find for each of the outcomes.

The calculator will then let you know whether an arbitrage opportunity exists, and how much you should stake on each of the outcomes to guarantee a profit.

It will also calculate the net profit for you.

Want access to a simple yet powerful arbitrage calculator?

Click below to take a look!

Of course, the other alternative is to use arb hunting software, which collects the odds data from tens of bookmakers on thousands of events and notifies you once an arb has been found.

It will also calculate the necessary amount to stake on each outcome.

Want to know which is the best value premium software for finding arbs?

Read my summary of arbitrage betting software!

Finishing up...

Whilst arbitrage betting formulas and calculations are fairly simple, they are very powerful as they allow you to turn regular gambling into virtually guaranteed profits.

In saying that, manually doing the calculations above for every market in every match with every bookmaker until you find an arbitrage opportunity can be very time consuming.

Even if you use the free online calculators, you still have to flick back and forth between browser tabs, manually entering the odds each time.

For those that are serious about sports arbitrage betting, purchasing a subscription to the arbitrage software is a no-brainer.

100% Free Video Course For Earning Money Online​With ​Arbitrage Betting

​The most comprehensive, in-depth training on sports arbitrage betting available. ​Start earning an income ​online using the unique techniques in this free course.

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Louis


I'm an Australian guy who has used sports arbitrage betting to provide a decent side income (thousands of dollars per month!) for myself while working full-time. I've set up the The Arb Academy to teach others how to do the same and achieve financial security through a second income stream!

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