Am I missing something? I reckon I can guarantee a financial win by betting on the federal election.

As you probably know, SportingBet Australia provide an election-betting service. For new accounts they are offering a sign-up bonus $100 free bet for initial deposits of $30 or more.

The conditions are here. Note conditions 5 and 6 in particular — only one free bet per event, and if your free bet wins you must pay back the $100 free stake-money.

The payout for a Labor win in the federal election is $1.85 for a $1 bet. The payout for a Liberal Coalition win is $1.95 for a $1 bet.

So imagine you signed up for $30 and put it on Labor and put the $100 free bet on Liberal Coalition.

If Labor wins, you make a profit of $25.50 (1.85 x $30 – $30). If Liberal Coalition wins, you make a profit of $65.50 (1.95 x $100 – $100 – $30). You have a GUARANTEED win of at least $25.50.

What is the maximum guaranteed win, and what is the corresponding bet? (Note this is not the usual question in gambling, which is how do I maximise my expectation?).

To simplify the problem, imagine I bet all of my initial deposit on Labor, and I bet all of my $100 free bet on Liberal Coalition. How much should I initially deposit and bet on Labor to maximise my guaranteed win?

Let Y be the amount I bet on Labor. Y must be at least $30 (condition 2). I place my $100 free bet on Liberal Coalition (I can’t split the bet according to condition 5). Assume a payout of $1.85 for a stake of $1 if Labor wins and a payout of $1.95 for Liberal Coalition.

Profit if Labor wins
win Labor = 0.85 x Y
Profit if Liberal Coalition wins
win Coalition = 95 – Y

Plot these on a chart where the horizontal-axis records the amount I bet on Labor and the vertical-axis is the amount I win. You will plot 2 straight lines. win Labor has a positive slope and passes through points (0,0) and (100,85). win Liberal has a negative slope and passes through points (0,95) and (95,0).

The guaranteed win is represented by the expression: guaranteed win = min (win Labor, win Coalition)

This expression is maximised at the intersection of the two straight lines you have just plotted. Recalling our simultaneous equations from high-school algebra, we can solve for Y as follows.

0.85 x Y = 95 – Y
Y = 95 / 1.85
Y = $51.35

So when I bet $51.35 on Labor and place my $100 free bet on Liberal Coalition I maximise my guaranteed win at $43.65.

win Labor = $43.65
win Coalition = $43.65
guaranteed win = $43.65.

Make money ($43.65) now. Ask me how.

Can this possibly be true? I assume SportingBet’s business model is predicated on gambling being addictive.