Mathematical expectation of a double chance bet

Math proof and calculations in the general case


The double chance betting strategy is an appreciable formula: by betting on two results at the same time from the same sporting event we very clearly and mathematically increase the probability of winning.
We win more often but, on the other hand, the gains are a little lower: lower risks but also lower gains...
Is this strategy clearly interesting? The mathematical answer to this question always fundamentally relies on the calculation of the mathematical expectation which is exactly the probability-weighted average.
In this article, we review these expectation calculations for a double chance bet, in the general case.
The result obtained is quite simple: a double chance betting strategy does not change the expectation in any way: in the long term, betting using a double betting strategy is the same as making single bets.
But this is a theoretical mathematical result relating to betting on the game of chance, only chance. The use in sports betting, with its specificities, remains ingenious, and the double chance bet is an essential tool and strategy to use systematically!


Double chance betting strategy

The double chance betting strategy simply consists of betting twice, on two results of the same event, and thus having two chances of winning instead of just one.
As only one of the two outcomes will occur, one of the two bets will certainly be lost, and the total gain will in any case be lower. On the other hand, you have twice the chance of winning, hence the name of the strategy and its interest.
The double chance calculator automatically calculates the amount of optimal bets to bet in a double chance bet, based on the odds of the outcomes.

In this article, below, we address the mathematical part and proof of this double chance betting strategy.
The main mathematical tool for studying a situation involving chance is mathematical expectation: that is what we will calculate, in the general case. We will prove that the mathematical expectation remains definitively zero (excluding the bookmaker's commission), using or not using a double chance bet.

Double chance expectation: math proof for the general case

We consider two different outcomes of an event on which we can bet, with the respective odds of c1 and c2, on which we bet the amounts m1 and m2.
We also know that the probability of winning each bet, separately and independently for each simple bet on these events, is the reciprocal of the corresponding odds, i.e. the probabilities p1 = 1/c1 and p2 = 1/c2.
This reciprocal relationship between the odds and the probability of winning is theoretical (and does not take into account, for example, the bookmaker's commission). See the fundamental article on odds and probabilities for more details.


Outcomes for a double chance bet, gains and probabilities

After a double chance bet, i.e. our two single bets on two different outcomes, there are three possibilities: We can now calculate the mathematical expectation, which is just an average weighted by the probabilities.
E = G1×p1 +G2×p2 + 0×pl
or also
E = m1×c1×1/c1 + m2×c2×1/c2 = m1 + m2
which shows that the expected value is exactly the stake involved, and thus that the expected net gain is zero.
We therefore proved that The double chance betting strategy does not change the expectation of a game of pure chance.

which is a particular case of the more general result, we do not reverse, by any strategy or formula, chance for our benefit!

But, as already mentioned : betting strategies (like double chance) are not useful in games of pure chance, but sports betting is not only about "pure chance" and so in sports betting , yes, (mathematical) strategies can be really effective in increasing your earnings.

Double chance bet: is it ultimately interesting?

The previous result shows the ineffectiveness of the double chance strategy. But it is, as already mentioned, ineffective against "pure chance", which is not the case with sports betting. Certainly, an element of chance remains in any sports bet, the vagaries of events, sporting surprises, ...
Any tipster can be wrong, and it is a fact, is regularly wrong: it is undeniable, no one can (and must not) guarantee 100% correct predictions.
On the other hand, if we systematically give two chances to a tipster, and even for an average tipster, it becomes very reasonable to (almost never) make a mistake. Especially when betting on events that are already relatively assured and low risk, with odds lower than 1.5 or 2.
Thus, a strategy such as double chance makes it possible to secure your bets, that is to say, to reduce the probability to lose.
In summary, we do win much more often (almost every time with serious predictions), but we win less each time.

Double chance is an essential strategy for ensuring long-term gains.


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