Odds, probability and margin

You don’t need to be a mathematician to catch arbs. One formula and one rule are enough. Five minutes.

Odds are just a multiplier

Put $100 on 1.95 and if it wins you get $195 back: your $100 plus $95 on top. That’s it.

But odds have a second meaning, and that’s the one we care about. Hidden inside them is the probability the venue gives that outcome.

probability = 1 / odds

odds    1 / odds
2.00    50.0%
1.50    66.7%
5.00    20.0%

The rule that shows an arb

Take the odds for every outcome, turn them into probabilities and add them up.

• Sum under 100%: there’s an arb. The profit is locked in.
• Sum over 100%: no arb, the venue took its cut.

Check it on the example from the previous page, 2.10 and 2.10:

1/2.10 + 1/2.10 = 47.6% + 47.6% = 95.2%
100% − 95.2% = 4.8%   ← that's your profit

95.2% is under 100%, so there’s an arb. The 4.8% between 95.2% and 100% is your profit.

Why it doesn’t work on one bookmaker

Because that’s exactly how bookmakers make money. A bookmaker sets its odds so the sum is over 100%. (If you’re curious, there’s plenty on YouTube about how and why the house always ends up ahead.) Same chances, but both prices inside one book:

1/1.90 + 1/1.90 = 52.6% + 52.6% = 105.2%
the extra 5.2% is the book's margin, its income

So you can’t build an arb inside a single bookmaker: there the probabilities always add up to more than 100%. An arb only appears between different venues, when their mistakes happen to line up in your favour. Catching that moment by hand is close to impossible, which is why you need something like Pulse watching every venue at once.