Even as a casual punter, you’d probably be aware of the availability of same-game multis which, put simply, allow you to construct a wager out of multiple “legs” (individual bets), usually up to some limit.

So, for example, you might multi “Geelong to win” with “First Goal Scored in First 2:00”. The price for the multi would usually be the multiplication of the price for the individual legs - say $1.80 (for Geelong to win) x $3 (for First Goal Scored in First 2:00) or $5.40, here - and the wager would only win if both of the legs are successful.

Not all same-game multis are allowed by bookmakers, however, and you might wonder why that is the case.

The answer requires us to understand and apply conditional probability.

THE SWANS AND BUDDY BAGFULLS

Let’s consider a prime example of a potential multi on the men’s AFL competition that would almost certainly not be allowed by bookmakers, and that would be to multi the following:

• Sydney to win, with

• Lance Franklin to kick 7+ goals

The core problem here is that those two wagers are likely to be highly positively correlated in that the event of Buddy kicking 7 or more goals is far more likely to be associated with a Swans’ win than with a Swans’ loss.

Let’s lay down some (theoretical) probabilities here (and let’s ignore the small probability of a draw):

• P(Swans win) = 40%

• P(Buddy kicks 7+ goals) = 5%

• P(Sydney wins given that Buddy kicks 7+ goals) = 90%

We see that the probability of Sydney winning is significantly increased - from 40% to 90% - if it is the case that Buddy kicks 7 or more goals.

Now, a fair price for a Swans win given a 40% probability would be $2.50, so a bookmaker who imposed a 5% vig would price them, instead at about $2.38. Similarly, a fair price for Buddy kicking 7+ goals given a 5% probability would be $20, and a bookmaker who imposed a 5% vig would price it, instead at about $19.05. (In reality, the bookmaker might impose a little more vig than this on such a relative longshot, but for simplicity’s sake I’ll just use 5%).

A standard Swans Win/Buddy kicks 7+ multi would then be priced at 2.38 x 19.05, or $45.35.

But, how likely is that compound event?

To answer that we need to apply some standard conditional probability theory for correlated events, which tells us that:

P(A and B) = P(A) x P(B given A).

In this case then, we have:

P(Buddy kicks 7+ goals and Sydney Wins) = P(Buddy Kicks 7+ goals) x P(Sydney Wins given Buddy Kicks 7+ goals)

which is 5% x 90%, or 4.5%.

The fair price for an outcome with a 4.5% probability is $22.22, but we saw earlier that the bookmaker would be willing to offer a price for the multi of $45.35 for this bet if he or she simply multiplied the individual prices for each leg, which implicitly means he or she is acting as if the two bets were uncorrelated.

Our expected return for every dollar wagered is then 4.5% x 45.35 = $2.04, or a profit of $1.04 per dollar wagered.

That’s a massively favourable wager, and one that no sane bookmaker is going to permit.

Which is, of course, why they don’t …

(or, if they do, will account for the correlation in the price they offer …

… as noted here by Sportsbet.)

So, the overall lesson here is that bookmakers will tend to preclude the ability to multi wagers where the outcomes are positively correlated - and, the more correlated they are, the less likely they are to allow them.

One other thing worth noting is that, where the legs of a multi are strongly negatively correlated, the price of the multi can represent extremely poor value.

Consider, for example, a similar similar scenario to the one above, but where we multi a Sydney loss with the Buddy bet.

The probabilities for this will be:

• P(Swans lose) = 60% (ie 1 - 40%)

• P(Buddy kicks 7+ goals) = 5%

• P(Sydney lose given that Buddy kicks 7+ goals) = 10% (ie 1 - 90%)

The fair price for the Swans loss is $1.67 which, with vig, will become $1.59. The price for the Buddy bet remains $19.05, so the price for this new multi will be 1.59 x 19.05 = $30.29.

But, the true probability is given by:

P(Buddy kicks 7+ goals and Sydney Lose) = P(Buddy Kicks 7+ goals) x P(Sydney Lose given Buddy Kicks 7+ goals)

which is 5% x 10% or 0.5%, meaning that the fair price is $200.

Our expected return for every dollar wagered here is 0.5% x 30.29 = $0.15, or a loss of $0.85 per dollar.

That is one hefty -EV.

Let’s finish by considering a multi where the probabilities for the two legs are uncorrelated.

Our new multi will be to pair a Collingwood Wins bet with a Total Score is Even bet (and we’ll assume that, in the event of a draw we just get our money back).

Say the probabilities for this are:

• P(Collingwood wins) = 75%

• P(Total Score is Even) = 50% (which is empirically true)

• P(Collingwood wins given Total Score is Even) = 75%

Note that the conditional probability here of a Pies win given an Even score is the same as the plain probability of a Pies win - knowing that the Total Score is Even gives us no new information about whether or not the Pies won and, conversely, knowing that the Pies won gives us no new information about whether or not the Total Score is Even.

Now the Pies win will be priced at something like $1.27, and the Even Score bet at $1.90, giving a price for the multi of 1.27 x 1.90 = $2.41.

The true probability is here given by:

P(Pies win and Total Score is Even) = P(Total Score is Even) x P(Pies win given Total Score is Even)

which is 50% x 75% or 37.5%, meaning that the fair price is $2.67.

So, instead of receiving $1.67 in profit for every dollar wagered, we’ll only get $1.41, and our expected return for every dollar invested is 37.5% x 2.41 = $0.904, or a loss of about $0.10 per dollar. By parlaying two wagers each with about 5% vig, we’ve created a multi with about a 10% vig.

That final result is generally true for all multis - same game or otherwise - that combine legs with zero correlation: essentially, all you are doing is multiplying the vig for the bookmaker.