Risk-free gambling with math!

Disclaimer: I do not endorse gambling and I have no idea if this is in breach of any gambling websites’ terms and conditions.

I’ve recently picked my first ever professional sporting team to support – the Boston Celtics. In doing so, I’ve finally discovered the allure of watching sports; once you assign value to a specific team winning, the game is far more exciting. This then led me to making $5 bets on games that didn’t involve my team, so I could get excited over other games that I watched with friends. After making a few bets on successive games, I quickly got bored and stopped visiting the gambling website. This is how I discovered how quick they are to offer you free bets. They knew deep down that I was a chronic gambler in the making; they just needed to keep me coming back for a while.

From free bets, you can only keep the winnings. For example: If you place a $10 free bet on something paying $1.10 for a win, then you have used up the free bet, and either have $1 or empty pockets. Assume, for the moment, that we’re dealing with a rather altruistic bookie. In this case, $1.10 for team A to win implies $11.0 for team B to win. In general, if we have $X for team A and $Y for team B, then they will be related by and the bookie’s odds are based on a probability of A winning, . Let’s try to understand these equations now.

If a bookie thinks that team A will win 10 out of 11 times, then he will expect to keep your $1 (that you’ve bet on team A) once in 11 games. This means that he can only afford to give you one tenth of a dollar each of the 10 times that you win; that is, he’ll offer you $1.10 odds for team A. On the other hand, if you bet on team B then he’ll expect to receive $10 from you over the 11 games. So now he can afford to pay you $10 on the time that you do win; that is, he’ll offer $11 for team B. So we could say that the odds are 1.10 to 11, in favour of team A, or team A will win 11 games for every 1.10 games that team B wins. (Note: This is the same as 10 games to team A for 1 games team B wins, but we’ve phrased it in terms of the known odds.) So we could say the probability of team A winning, is , which hopefully explains the formula given above for .

Sadly I don’t know any altruistic bookies, so we’d better take their cut into account! Let’s imagine the same scenario, but now the bookie is keeping 5% of the money that comes in. Over the 11 games (and $11), the bookie will pocket $0.55 and so the $1 you would have previously won is now only $0.45. Since this $0.45 is to be distributed over 10 games, the odds he would give you for team A is now only $1.045. If you are betting on team B, he will still keep his $0.55 but this will come out of the $10 that he could afford to pay you for winning. This means the odds for team B would be given as $10.45. If Z is to be the bookie’s cut (in this case Z=0.05), then the astute reader should be able to verify the following relationship between X and Y: . Note: The probabilities and are given by the same formulas.

Now, lets go back to that free bet I was talking about. Say we place a $100 free bet on team A to win, with odds given by $X. If team A wins, then we get $100(X-1) in our pocket and if team B wins then we’ve lost our free bet. Let’s now imagine that we bet something less than $100(X-1) on team B, so that if team A wins then we’re not losing anything and if team B wins then we’ve won our real bet and lost the free bet. It is actually possible to judiciously choose our bets in such a way that can be assured $63 cash from our free bet!

If we put an $a free bet on team A and $b of our own money on team B, then we have two possible outcomes:

• Team A wins: You gain $((X-1)a-b)
• Team B wins: You gain $(Yb-b)

Note that we have subtracted the $b that we’ve invested.
For now we are interested in a guaranteed win, so we would like both outcomes to result in the same gain for us. That is, we need . Now either way, we gain . But we have a relationship between X and Y from before, so we can write this in terms of the single quantity Y.
Total gain .
At this point we will simply state that the maximum total gain occurs when and results in a gain of – the reader who remembers their calculus can easily verify this.

This is starting to get a little messy now, but using these equations with Z=0.04 (seems about right for the betting website I found), you can guarantee 63 cents on every dollar of free bets they offer. Simply look for a game with odds around $1.19 against $4.90; place your free bet on the team paying $4.90 to win, and place a bet 3.27 times larger than the free bet on the $1.19 team.