Surebet: a complete mathematical review and proof

surebet refers to a betting strategy where you place bets on all possible outcomes of a sporting event, guaranteeing a profit regardless of the actual result.
This is not a tip to be taken at one's word, but is the result of a clear and complete mathematical calculation.
Let us finally add, before going into details, that it is not a betting strategy which guarantees an unconditionally profit in any situation, but which aims to take advantage of a particular situation, which, in turn, is not necessarily systematic.
Below we explore the mathematical details, algebraic calculations, that is the whole maths proof.

Surebet in sports betting could be compared to an eclipse phenomenon at a bookmaker or between several bookmakers. That's a rare and exciting phenomenon, but one have to know when and where to look to see it…
Same way, appearance of a possibility of a surebet is quite improbable for reasons that we will detail later.
However, improbable does not mean non-existent, especially for a wagerer who plays fairly and wishes to triumph over apparent randomness and probabilities ! Furthermore, understanding this mathematic principle is an essential step towards a deeper understanding of how sports betting works.
Furthermore, even if "natural" surebet situations are rare, it is possible to provoke them: we will thus end up with secure sports betting formulas (that is to say not winning for sure, but with a much reduced risk of loss, see the Double Chance formulas or Refunded If or even Math tip: multiple bets secured using surebet.

The recipe (mathematical and exact as we will see and prove it) is as follows: find the implied probability of each outcome (which can be found by taking the reciprocal of the odds), then add these implied probabilities of every outcome. If this resulting sum is less than 1, here is a surebet opportunity.
The aim of what follows is twofold: to clarify, justify and mathematically prove this formula, and also to specify how to calculate how much to bet on each outcome to garantee the maximum profit.

Surebet example on a two way event: win or loss - details of Math calculations

To explore the general situation, let's consider the event with two outcomes: victory or defeat for which we denote Total stakes are then mt = mv + md. We the calculate profits for each outcome, that is the alternative I now look for bets optimizing my winnings, for sure, that is to say regardless of the result of victory or defeat.
Firstly, if one of the two profits Gv or Gd is greater than the other, my strategy will not be optimal, because I would have a chance of winning only the smaller of the two gains.
A first condition is therefore to have equality of these net gains, i.e.
Gv = Gdmv×cvmt = md×cdmtmv×cv = md×cd
We thus find the optimal mathematical relationship between the two bets on victory and defeat:
md = mv cv/cd
We then calculate the expression of the guaranteed profit,
G = Gv = Gd = mv×cvmt = mv×cvmv md
and therefore replacing the stake md inside previous formula
G = mv×cvmv mv cv / cd
and finally, with some algebraic calculations, and factorization by mv×cv, we obtain
G = mv×cv ×(1 − 1/cv1/cd)

Afetr all, this last relationship is only interesting if, in addition to the equality of the two gains, these ones are positive (otherwise we are guaranteed to lose, an identical loss in both cases, for sure...), which implies that
G = Gv = Gd > 0
and so here,
(1 − 1/cv1/cd) > 0
and we find the mathematical relationship characteristic of surebet opportunities:
1/cv + 1/cd < 1

Example 1: without surebet opportunity

Let's consider the sport event for which We then calculate the mathematical expression characteristic of the surebet opportunities:
1/cv + 1/cd = 1/1,15 + 1/2,2 ≃ 1,3 > 1
and we don't find any surebet opportunity here: it's just a risky bet…

Example 2: with surebet opportunity

Let's now consider the sport event for which We can compute this time that the surebet opportunities is
1/cv + 1/cd = 1/1,30 + 1/5,00 ≃ 0,97 < 1
which shows a surebet opportunity.
Our previous calculations then give us the distribution of bets (how much to bet on each outcome), for example for a bet on victory of mv = 100 euros, we then bet
md = mv cv/cd = 100×1,30/5,00 = 26
that is 26 euros.
The total stake is therefore 126 euros and we then check that Whatever the outcome, a 4 euros net profit is guaranteed. If we can afford it, we might as well wager 1000 euros on victory and 260 on defeat, for a guaranteed gain of 40 euros, or why not wagers of 10,000 euros and 2600 euros for a guaranteed net gain of 400 euros ??

That is somehow low profit compared to stakes. The mathematical tool for evaluating the quality of an investment is the "ROI", which stands for Return On nvestment, and which is defined by the math formula

ROI = Net profit/Total bet
In previous example, this ROI is therefore
ROI = 4/126 ≃ 3%
which is somehow low … but, once again, guaranteed…

See the surebet calculator Lien to automatically compute surebet opportunities and, if so, the optimum distribution of bets.

Simplified surebet detection formula

The previous mathematical characterization of surebet opportinities is not so easy to handle: computing the reciprocal of decimal numbers in your head is not given to everyone… an automatic tool is welcomed, like this one Lien.

What ultimately interests us is actually the net profit. To obtain this one, we must subtract our stake to the calculation of profit with odds which.

Thus, if the odds for victory are cv, we can explode this coefficient as
cv = 1 + tv
where cv is the rate giving precisely the net profit.
For example, with odds of cv = 1,60, if I bet 50 euros and win, I earn 50×cv = 80 euros, but my net profit is only 50×0,60 = 30 euros.
Here we have cv = 1,60 = 1 + tv, with the "net odds" tv = 0,60.
In the study explored here, with odds for victory cv = 1 + tv and odds for defeat cd = 1 + td there are some surebet opportunities when
tv × td > 1

This condition is mathematically equivalent to that on the sum of the reciprocals of the odds (or implies probabilities) (see complete and detailled mathematical proof).
This formula is moreover simplier to handle, even without calculator.
Example 1 above can be written, according to this new formula, We then have, much more easily, the calculation
tv × td = 0,15 × 1,2 = 0,18 < 1
and there is therefore no surebet opportunity.

Concerning preceding example 2, using net profits and odds formulations, We the find, again, but much more easily, with the simplified math computation
tv × td = 0,30 × 4 = 1,2 > 1
there are here some surebet opportunities .

Surebet for a three-way 1x2 sport event (or victory/draw/defeat)

Three-way betting odds offer three wagering options. They differ from two-way odds as a TIE is added as a third betting choice. Three-way lines are offered in most competitions where a draw is a possible outcome, such as for the popular soccer betting options.
So let's now turn to these three-way sport events and betting. Three ways are victory / draw / defeat, for which we assume to have Total stakes are thus mt = mv + mn + md.
We the calculate profits for each outcome, that is the alternative I now look for bets optimizing my earnings, for sure, that is to say regardless of the outcome victory, draw or defeat.
Firstly, if one of the three profits Gv, Gn or Gd is greater than others, my strategy will not be optimal, because I would have a chance of winning only the smaller of the three gains.
A first condition is therefore to have these profits equal, i.e.
Gv = Gn = Gd
These equalities directly lead to the optimal mathematical relationships between the three bets:
mv×cv = mn×cn = md×cd
The guaranteed profit is thus
G = Gv = Gn = Gd = mv×cvmt = mv×cvmvmn md
and then, replacing the bets mn and md inside the previous optimality formula
G = mv×cvmv mv cv / cn mv cv / cd
and finally, with a some algebraic calculations, and a factorization by mv×cv, we obtain
G = mv×cv ×(1 − 1/cv1/cn1/cd)

Finally, this relationship is only interesting if, in addition to the equality of the three gains, theses are positives (otherwise we are guaranteed to lose, an identical loss in all three cases, for sure...) , that is
G = Gv = Gn = Gd > 0
and so here,
(1 − 1/cv1/cn1/cd) > 0
and we finally find the mathematical relationship characteristic of surebet opportunities:
1/cv + 1/cn + 1/cd < 1
and we arrive at the same mathematical relationship characteristic of surebet opportunities for two outcomes.
This surebet opportinities characterization is not so easy to handle, but there is not any simplified form this time. You can use the surebet automatic calculator which will automatically compute and yield surebet opportunities and, if so, the optimum distribution of bets as well as the ROI.

How and where to find surebet opportunities

As we said, surebet opportunities appears in a particular situations. These situations should not be found inside a single bookmaker offers: this would mean that, in this particular event, the bookmaker puts himself in a situation of guaranteed loss ! unless this is a particular calculation or verification error on his part... therefore to be exploited quickly if it exists, before the bookmaker detects and corrects it !

Surebet opportunities are therefore to be looked for between several bookmakers, opportunities resulting from different estimations of the outcome of an event: one will estimate, for example, the victory of a player or a team much more probable than the other, and odds offered by these two bookmakers will be quite different endind with a possible surebet opportunity.
To evaluate odds, bookmakers also take into account, among many elements and statisical data, odds offered by their competitors and thus tend a kind of harmonization: a big divergence between odds from one bookmaker to another rarely occurs ! for example, a team which is only underdog for a bookmaker is also underdog for the other bookmaker. This is mainly true for the most followed events, with the highest betting volume. On the other hand, for less popular competitions, league 2 or equivalent in other countries, or for example in tennis, tournaments other than the major Grand Slams (Masters 1000, or ATP 500 or 250), are much more likely to offer odds quitly differing and sometimes leading to surebet opportunities.
Following the same idea, one ca also pay attention to less publicized events, therefore with lower volumes of bets, for example other sports such as hockey, volleyball, or even handball… These will therefore also be less rigorously scrutinized, therefore more likely to present cases of surebet.

Why not get (too) tired of searching for surebets opportunities ?

To find surebet opportunites, at least 2 bookmakers are thus required. Searching in a large number of bookmaker offers quickly turns out to be tedious, having to go through all odds, looking for odds that meet the surebet opportunity mathematical formula.
Task can be cut short by only looking for surebet opportunities for two-way events. The formula is already simpler in this case, especially in the simplified version.
Paid sites and services exist, but as the profit with surebet betting is low, also adding (or instead substracting…) a commission from an intermediary... Finally, in practice, here is a list of reasons why it is not really wise to tire yourself out searching for and exploiting surebet opportunities:

Did I then read, understand and calculate everything for nothing ?

No way ! Overall, understanding all the calculations, maths, the advantages, drawbacks, etc... around surebets, is clearly an important progress in the knowledge of the betting mechanisms. In particular, it is now possible to calmly tackle other betting elements, and to come around profitable strategies:

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