Math expectation of different sports betting formulas

Betting strategy do not change expectation: math proof

Mathematical expectation is the mathematical calculation tool which quantifies the profitability, or not, of a risky bet, sports betting in particular.
By relying purely on chance (without external information, tipster, stats, etc.) this expectation is in theory zero (this is where the very calculation of the odds comes from), and even in practice negative because intermediaries like bookmakers take additionally their commission.
For this reason, everyone is looking for winning betting strategies and formulas, that is to say whose expectation is positive. For example, the most well-known and common betting strategies: In this article, we will see that none of these methods actually changes the math expectation: whatever the way of betting, the expectation of the simple game is preserved; in the long run, that is a significant number of bets, the total winnings will remain approximately the same.

Expectation for the classic truncated martingale

In the classic martingale series of bets you repeatedly double your bet until you win. That is theory, actually, whatever your betting capital, your bankroll, no one can bear an infinite series of losses.
We thus initiate our martingale with for example 1 euro and double our stake with each loss.
With a single game, we have a 1 in 2 chance of winning 1 euro and a 1 in 2 chance of losing our euros (so odds = 2). The expectation is simply zero: over a large number of games, our average gain will be approximately zero, no gain or loss either.
Let's come back to the martingale and say that we have a bankroll of 10 euros: beginning with a 1 euro bet we therefore cannot lose more than 3 times in a row, i.e. the loss of 1+2+4 = 7 euros, and this happens with probability p = 1/23 = 1/8 . In all other cases, we earn 1 euro (net profit).
We therefore have the table of gains and associated probabilities (which is actually the mathematical law of probability):
and we then easily calculate the expectation:
E = 1×7/8 + (−7)×1/8 = 0
and hope and therefore the same with the martingale as by simply betting once.

Now, not being able to afford more than 3 consecutive losses on a martingale is a real mistake for the bettor (read more about this subject in diversify your strategies !; it is absurd to bet 10% of your bankroll in a single bet…).
The probability of losing a martingale decreases more clearly with the number of rounds, so we can think that if we can support more loss rounds in the martingale, chances to get the final profit increase, and the same for the strategy expectation. It is simply not so !
Let's say for example now that we can afford to lose 10 times in a row on our doubling martingale. The probability of such a loss is now
p = 1/210 ≃ 0.001 = 0.1%
This probability is very low, but the loss in this case is also more significant:
1 + 2 + 4 + … + 512 = 1023 ≃ 1000 euros
and the new table of gains and associated probabilities (law of probability):
and we calculate the new math expectation:
E = 1×0.999 + (−1000)×0.001 = 0
which shows that expectation still remains zero.

These two calculations can be generalized: whatever the bankroll and the number of consecutive losses that we ca afford, the math expectation remains definitively zero !
Setting up a martingale strategy does not change the expectation, that is to say the long-term gain.

It doesn't all amount to the same thing though. The standard deviation, on the other hand, changes depending on the available bankroll: setting up a martingale does not change the expectation but does change the risk: we play higher stakes using martingale system.

Combined bets

Another betting startegy: the combined bet which consists of grouping several bets into a single one. The odds of the bets that compose the combined bet are then multiplied together, which gives an attractive overall odds. For example, let's bet 10 euros on the combination of 3 bets: If I win, I earn 10×1.5×1.8×1.7=45.9 euros, that is 35,9 euros net profit, with the attractive overall odds of 1.5×1.8×1.7=4.59.
The winning probabilities of these 3 bets are respectively 1/1.5≃66%, 1/1.8≃55%, and 1/1.7≃59%, and the probability of winning the combined bets, therefore the 3 bets simultaneously is therefore approximately 66%×55%×59% ≃ 21% .
Same way as before, we sum up in the table of gains and probabilities:
and calculate the math expectation
E = −10×79% + 35,9×21% ≃ 0
and expectation remains zero.
In other words, combining bets does not change the overall expectation but changes the distribution of winnings. By betting on bets separately and independently the winnings will be lower on average, but more frequent than in the combined bets. The calculation of the expectation shows that this change in distribution won't be profitable in the long term.
Combined bets are lures that stimulate the desire to bet with more attractive odds. "stimulate" the desire to bet ? this is the job of bookmakers… (who charge, which is not counted in the previous calculations, an additional commission inside their combined bets offers).

As with martingales, combined bets does not change the expected value, i.e. the long-term gain.
Only the risk evolves and, with possible significant gains (but therefore, mathematically, exactly compensated in the long term by equivalent losses).
This kind of formula is therefore a first choice tool for those seeking to encourage bettors, some marketing tool: high stakes and big gains in perspective (without risk for the bookmaker)

Refunded if, or Draw no bet, betting strategy

Another betting strategy: the "Refunded if" mathematical strategy, or "Draw no bet" strategy, which apparently "secure a bet" by distributing the overall stake over two outcomes.
For example, I want to bet 10 euros on my favorite team which will win, for sure ! well, actually, almost sure... well let's just say that I wouldn't like to see a draw that's not so improbable...
The odds on this match are: 1.9 for victory and 4 for a draw.
What to do ?
Using the calculator I find that I can distribute my 10 euros stake according to: The fear of a "not so improbable" draw disappears, I no longer have anything to fear.
Let's look at the payout table and corresponding probabilities
and the resulting math expectation
E = 4.25×53%+0×25% −10×22% ≃ 0
and the expectation remains zero just as for the classic bet without Refunded if strategy where the payoff table is
and same zero expectation:
E = 9×53% −10×47% ≃ 0

Again this startegy Refunded if does not change the expectation, which remains zero (even less the margins of intermediaries). We only redistribute the gains in order to mitigate the risks. It's actually the standard deviation that's decreasing here.
In the long term, reiterating Refunded if bet strategy does not change the average winnings.

As with the previous strategies, the Refunded if strategy does not affect the mathematical expectation and therefore the long-term profits.
On the other hand, the risk (the standard deviation, mathematically) is well attenuated, hence the short-term interest of the formula.

double chance

The double chance strategy works a bit like the Refunded if strategy, except that you do not only seek to your stake back on a second outcome, but to be moreover also winner in this case.
Using the example used for the above refunded if strategy in the previous paragraph, and using the double chance calculator we find that we can distribute the 10 euros stake according to: So, here I am a winner for the victory and the draw, with a net profit of 2.88 euros.
The payout and odds table is now
and the math expectation
E = 2.88×78% −10×22% ≃ 0
remains zero.
Expectation of gain in the long term remains still the same, even if the startegy sounds attractive: in the distribution of gains, we have transformed part of the losses into gains! but in return this gain is lower... and this will be exactly compensated in the long term...

As with the previous strategies, the Double chance strategy also does not affect the math expectation and therefore the long-term winnings.
On the other hand, the risk is further reduced here compared to the Refunded if strategy, hence again the interest, therefore in the short term, of the formula.

See this page for a full mathematical review and proof about double chance expected value.

Trixie betting strategy

Finally, the trixie betting strategy is a very attractive alternative to the treble combined bet which is normally not recommended not recommended due to this high risk level (and we saw here in addition that it brought nothing in the long term, its expectation remaining zero…).
A trixie bet is a kind of hybrid strategy: it is a treble bet also along with a kind of securing formula Refunded if: By correctly adapting the 4 stakes, which can be automatically done by this calculator, I can ensure that I win the jackpot if my treble is correct, and refund my overall bet if one of the three bets is lost (so the treble bet is lost).
The calculator yields optimized stakes calculation results and corresponding winnings and probabilities.
Once again, and this is the aim of the "Refunded if" part, we transfer part of the losses into gains, by mitigating in return the main gain of the triple combination.
The calculation of the math expectation then shows that, systematically, it remains zero (and the various margins of intermediaries would still have to be removed...).

As usual now, this betting strategy does not affect the expectation and does not contribute in any way in the long term profit, but reduces the short term risk taken (mathématically, the standard deviation).

See this page for a full mathematical review and proof about trixie expected value.


We have all along this article talked about pure chance games. In this context, there is no strategy that has any effect on the math expectation: the odds are made precisely so that the expectation is zero (negative, actually, because of bookmakers margin).
This general result, namely "no method of betting changes the long-term expectation of a game of chance", has been well studied by Richard Arnold Epstein, a specialist in game theory, in "The Theory of Gambling and Statistical Logic":

"If a gambler risks a finite capital over many plays in a game with constant single-trial probability of winning, losing, and tying, then any and all betting systems lead ultimately to the same value of mathematical expectation of gain per unit amount wagered."

In summary, we do not overturn, by any strategy or formula, chance for our own benefit !

In summary again, all the strategies, systems, formulas, ... allow you to adapt your way of betting, the search for a mathematical formula allowing you to win every time, or in the long term, is illusory !
All is not hopeless, however, because sports betting is not only a game of pure chance, and a lot of information allows predictions to be directed towards value bets which are precisely by definition bets with positive expectation, i.e. - say winners in the long term.
There are therefore many betting strategies, methods and formulas in sports betting which make it possible to ensure or secure your bets and increase your winnings.
For example, system 2 of 3, if it does not deviate from the rule by not modifying the expectation of winning, clearly ensures winning much more often. Likewise, this math betting tip on multiple bets completely illustrates all these points again: the expectation is still not modified, but the betting strategy allows you to fully use safe value bets by increasing their odds and clearly reducing the probability of loss, making predictions much easier, even less precise.

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