Surebet: a simple calculation formula

A surebet is a bet where you are sure to win regardless of the actual outcome.
This can be mathematically possible by placing bets on all the opposing outcomes of a contest (possibly with different bookmakers).
Of course the possibility of surebet is not systematic, and is even a rather rare opportunity. You have to know how to detect it.

Classic formula

The classically known calculation formula is as follows: in a win/loss bet the sum of the reciprocal of the odds must be less than 1. (on this page are the explanations and details of the mathematical calculations).

Win with odds of c_v = 1.30
Defeat with odds of c_d = 5.00

We then have

1/c_v + 1/c_d = 1/1.30 + 1/5.00 ≃ 0.97 < 1

and there is a little room for a surebet there.
Although easy, these calculations are not done in your head, and therefore you always need to have a calculator in your pocket, or you use an online automatic calculator or 3-ways 1x2 type calculator.

Simple formula

We prefer the much simpler form, using net odds or net winning rates

t_v t_d > 1

I n previous example, t_v = 0.30 et t_v = 4 and then it is much simpler to calculate

t_v t_d = 0.30×4 = 1.2 > 1

which shows in the same way that we are in a surebet situation.
Subsequently, below, we specify and prove this math formula.

Math proof of the simplified formula

The simplified formula is not a completely "magic" formula. It is obtained mathematically, like the classic formula. This is how, for fans of algebraic mathematical calculation (or those who would like to delve into it).

The odds give the gross profit. What interests the bettor is rather the net profit: the gross profit from which we withdraw the stake which is ultimately not part of the profit.
Thus, if we bet 10 euros with odds of 1.30, the gross profit is 10×1.30=13 euros, while we have actually won "only" 3 = 13 - 10 euros.

To highlight the net winnings, we write the odds in the form

Victory with odds of c_v = 1 + t_v
Defeat with odds of c_d = 1+ t_d

In the previous example, we have for example odds of c_v = 1.30 = 1 + 0.30 and net odds of t_v = 0.30 = 30%.
Now, the previous formula is rewritten, writting the fractions on the same denominator Maintenant, la formule précédente se réécrit, en mettant sur le même dénominateur les factions

1/c_v + 1/c_d <1 ⇔ 1/1 + t_v + 1/1 + t_d <1 ⇔ 2 + t_d + t_v/(1 + t_v)(1 + t_d) <1 ⇔ 2 + t_d + t_v < (1 + t_v)(1 + t_d)

Finally as for the right hand side, by developing:

(1 + t_v)(1 + t_d) = 1 + t_v + t_d + t_v t_d

and the previous inequality becomes more simply

t_v t_d > 1

formula which is much simpler to use, even without a calculator.

Thanks to this formula, we can quickly detect, even approximately in a first step, a situation which could be a surebet, then only in a second step draw out our surebet automatic calculator to check and calculate the right stakes.

Example

We consider the event:

Victory with odds of c_v = 1.40 = 1 + 0.4
Defeat with odds of c_d = 4.00 = 1+ 3

and with the formula, we calculate directly, in our heads, that

0.4×3 = 1.2 > 1

There is therefore a surebet opportunity, and we can now turn to the calculator which will tell us that if we want to wager, say, 10 euros in total, we must distribute them as follows: 7.40 euros on the victory, and 2.60 euros on the defeat , and we will have a guaranteed gain of 5 euros (excluding bookmaker commission...)