# Trixie bet expected value

## Math proof and calculations in the general case

The trixie betting strategy is known to be an attractive betting strategy with potential high payouts, while minimizing the risks.

The same fundamental question still here arises: how these "high payout" and "low risks", that is low probability of loss, balance each other ?

The mathematical answer to this question relies on the calculation of expected value which is exactly a probability-weighted average.

In this article, we go through these calculations, in the general case.

The result is quite simple: a trixie betting strategy do not change, at all, the expected value: on the long-term, wagering using a trixie strategy is the same as making simple bets.

The same fundamental question still here arises: how these "high payout" and "low risks", that is low probability of loss, balance each other ?

The mathematical answer to this question relies on the calculation of expected value which is exactly a probability-weighted average.

In this article, we go through these calculations, in the general case.

The result is quite simple: a trixie betting strategy do not change, at all, the expected value: on the long-term, wagering using a trixie strategy is the same as making simple bets.

## Trixie betting strategy

Here we consider three different events that we can bet on. A trixie bet is a multiple bet which includes three doubles (two selections combined) and one treble (all three selections combined). The treble yields a high potential profit thanks to a combined odds (multiplication of odds), while the three double bets are securing and minimizing chance of loss (same idea as in the Refunded if (or Draw no bet) strategy)). In an accumulation bet, like a treble, only a single mistake and all is lost. With a trixie on the opposite, even in the event of an error, the secondary double bets take over while still providing a gain.In this article, below, we address the mathematical part and proof of this trixie 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, using or not using a Trixie bet.

## Trixie expectation: math proof for the general case

We consider three different events, with respective odds of*c*

_{1},

*c*

_{2}and

*c*

_{3}. We know moreover that probability of winning is, for each single bet on these events, the reciprocal of corresponding odds, that is respectively

*p*

_{1}= 1

*c*

_{1},

*p*

_{2}= 1

*c*

_{2}, and

*p*

_{3}= 1

*c*

_{3}. This reciprocal relationship is theoretical (and do not take for example into account the bookmaker's commission). Read the fundamental article on odds and probability for more details.

### The four bets of the trixie, winnings and probability

In a trixie bet, we place 4 bets:- We bet an amount
*b*on a treble, accumulating the three bets of the three events._{t}

The odds of this treble is the multiplication*C*=_{t}*c*_{1}*c*_{2}*c*_{3}, and the probability of winnings is also the multiplication of each corresponding probability:*P*=_{t}*p*_{1}*p*_{2}*p*_{3}= 1*c*_{1}×1*c*_{2}×1*c*_{3}=1*c*_{1}*c*_{2}*c*_{3}*G*=_{t}*b*×_{t}*C*_{t} - We place 3 double bets, on events 1&2, 1&3, and 2&3. The amount of each bet is denoted respectively
*b*_{12},*b*_{13}, and*b*_{23}.

To win only the double bet 1&2, we must win bet #1**and**bet #2**and not**bet #3.

Bet #1 is won with probability*p*_{1}; bet #2 is won with probability*p*_{2}, while bet #3 is lost with probability (1−*p*_{3}),

The probability of winning for this double bet the is thus the product*P*_{12}=*p*_{1}*p*_{2}(1−*p*_{3}), with the double bet gain of*G*_{12}=*b*_{12}*c*_{1}*c*_{2}

The same applies for double bet 1&3, with the corresponding probability of winning*P*_{13}=*p*_{1}*p*_{3}(1−*p*_{2}) and the potential winning of*G*_{13}=*b*_{13}*c*_{1}*c*_{3}

and again for double bet 2&3, with the probability*P*_{23}=*p*_{2}*p*_{3}(1−*p*_{1}). and gain*G*_{23}=*b*_{23}*c*_{2}*c*_{3}

### All the trixie outcomes, winnings and probability

We are now in a five-case scenario:**All bets are won:**we win in this case the treble bet and also all three double bets. Total Gain isThe probability of this outcome is the multiplication of each single outcome probability, that is*G*=*G*+_{t}*G*_{12}+*G*_{13}+*G*_{23}*P*=_{t}*p*_{1}*p*_{2}*p*_{3}= 1*c*_{1}×1*c*_{2}×1*c*_{3}=1*c*_{1}*c*_{2}*c*_{3}**win bets #1 and #2 and bet #3 fails:**in this case, we lose the treble bet and also the two double bets*b*_{13}and*b*_{23}, but win the double bet*b*_{12}with the gain*G*_{12}.

This case happens with probability*P*_{12}**win bets #1 and #3 and bet #2 fails:**in this case, we also lose the treble bet and two double bets:*b*_{12}and*b*_{23}, but win the double bet*b*_{13}with the gain*G*_{13}.

This case happens with probability*P*_{13}**win bets #2 and #3 and bet #1 fails:**in this case again we lose the treble bet and the two double bets*b*_{12}and*b*_{13}, but win the double bet*b*_{23}with the gain*G*_{23}.

This case happens with probability*P*_{23}**Lose two bets or all three bets:**in these last cases, we simply lose all bets, gain is nul.

This happens with probability*P*which is not really relevant to calculate because, in the expected value calculation, it will be multiplied by 0: the associated gain._{l}

### Math expected value calculation

We are now in position to calculate the expression of the expected value, which is then the probability-weighted average:*E*=

*G*×

*P*+

_{t}*G*

_{12}×

*P*

_{12}+

*G*

_{13}×

*P*

_{13}+

*G*

_{23}×

*P*

_{23}+ 0×

*P*

_{l}

*G*×*P*= (_{t}*G*+_{t}*G*_{12}+*G*_{13}+*G*_{23})*p*_{1}*p*_{2}*p*_{3}= (*b*_{t}*c*_{1}*c*_{2}*c*_{3}+*b*_{12}*c*_{1}*c*_{2}+*b*_{13}*c*_{1}*c*_{3}+*b*_{23}*c*_{2}*c*_{3})1*c*_{1}*c*_{2}*c*_{3}=*b*+_{t}*b*_{12}1*c*_{3}+*b*_{13}1*c*_{2}+*b*_{23}1*c*_{1}=*b*+_{t}*b*_{12}*p*_{3}+*b*_{13}*p*_{2}+*b*_{23}*p*_{1}-
*G*_{12}×*P*_{12}=*b*_{12}×*c*_{1}*c*_{2}×*p*_{1}*p*_{2}(1−*p*_{3}) =*b*_{12}×*c*_{1}*c*_{2}× 1*c*_{1}×1*c*_{2}(1−*p*_{3}) =*b*_{12}(1−*p*_{3}) - likewise, we get
*G*_{13}×*P*_{13}=*b*_{13}(1−*p*_{2}) and*G*_{23}×*P*_{23}=*b*_{23}(1−*p*_{1})

*E*=

*b*+

_{t}*b*

_{12}

*p*

_{3}+

*b*

_{13}

*p*

_{2}+

*b*

_{23}

*p*

_{1}+

*b*

_{12}(1−

*p*

_{3}) +

*b*

_{13}(1−

*p*

_{2}) +

*b*

_{23}(1−

*p*

_{1})

*E*=

*b*+

_{t}*b*

_{12}+

*b*

_{13}+

*b*

_{23}

This result shows that the net profit expectation is nul.

We have proven here that The trixie strategy does not change the expected value of a game of pure chance.

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

But, as we already mentioned: mathematical betting strategies (like trixie) are really not useful in games of pure chance,

**but sports betting is not "pure chance"**and thus in betting sports, yes, mathematical strategies can really be useful for winning.

## Trixie bet: is it worth it?

The previous result shows the ineffectiveness of the Trixie 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, etc.Any tipster can be wrong, and it is a fact, they are wrong regularly: it is undeniable, no one can guarantee 100% correct predictions.

On the other hand, on the predictions of 3 events, and even for an average forecaster, it is unthinkable to be wrong 3 times out of 3! or we can clearly question our tipster for the future). Especially when betting on low-risk events, with odds less than 1.5 or 2.

Thus, a strategy such as trixie makes it possible to secure your bets, that is to say to avoid losing: in the event of an error, the loss is low (or even zero, with the particular case of trixie with odds of 2).

In summary, we don't necessarily win every time, but we don't lose. So, in the long term, we are simply a winner...

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