New York State legalized mobile sports gambling in January 2022. The casinos offered crazy promotions to get people signed up. The craziest offered $3300 of free money. I did the math, and there was a low risk way to claim this money. As long as you placed 11 or more bets, you had less than a 1% chance of losing the money you put in, but on average will walk away with an extra $3000. If you are interested in the probabilities of winning/losing, scroll down to "The Math". Unfortunately, this promotion ended on January 21st, 2022.

I generally think gambling is a waste of money and time. However, working on the probabilities here was pretty interesting, and it worked out for me. I'm also not the only person who thinks this was basically free money. See What the fudge is happening with NY sports betting promos? for a second point of view. I'm pretty sure this promotion used to run until January 31st, so the casino eventually figured out that they did the math wrong on this one and cut it short.

The deal was open an account, deposit $3000, and you got $3300 of "bonus cash", which converts to real money after you bet it. This means you needed to bet the entire $6300, $3000 of our own money, and $3300 of the casino's money. We get to keep whatever winnings come out the other end. My goal was to get the casino's money, without losing my own. The strategy was to place a large number of bets that are highly likely to happen. This minimizes the chance of losing, and decreases the variability in outcomes. The downside is the most you could take home is limited to approximately 1.5x the initial bet ($9450). (An even safer strategy is to bet both sides of the event, but that is forbidden by the terms of the promotion, see below.)

Each bet has "odds" that the casinos adjust to ensure they have a slight edge, but they try to stay "neutral". For my model, I assume that the casino's odds are an accurate assessment of the true probability of the event. This is not completely accurate. The odds represent the betting activity, but the "wisdom of the crowds" theory suggests the casino's odds should be close, particularly for popular events. For example, a -200 bet has implied probability of 66.7% of winning, and pays out 1.5x the initial bet to balance the payoff and risk. (See American Odds for a description and Odds Converter for a calculator.) The following model assumes that every bet uses this probability and payoff (-200, 66%, 1.5x). A better model would take into account the "house edge", which is around ~4-5% [1] [2].

If you were to place a single bet, then 66% of the time, you win 1.5x your money, but 34% of the time, you lose it all. On average, this strategy pays out the money that you put in. However, it has a 34% chance you lose it all, which I don't like. To reduce that probability, we make more uncorrelated bets. The more bets we make, the more likely it is that we won't lose them all.

The following table shows that the expected profit for the number of bets does not change, but the probability of losing money (the last column), generally decreases as the number of bets increases. For details on how this was calculated, see the Python script I used to calculate this.

num bets | expected profit ($) | prob. of losing money |
---|---|---|

1 | 3237.00 | 0.340000 |

2 | 3237.00 | 0.115600 |

3 | 3237.00 | 0.039304 |

4 | 3237.00 | 0.117126 |

5 | 3237.00 | 0.048643 |

6 | 3237.00 | 0.019537 |

7 | 3237.00 | 0.049225 |

8 | 3237.00 | 0.021793 |

9 | 3237.00 | 0.009358 |

10 | 3237.00 | 0.022042 |

11 | 3237.00 | 0.010083 |

12 | 3237.00 | 0.004492 |

13 | 3237.00 | 0.010195 |

14 | 3237.00 | 0.004758 |

15 | 3237.00 | 0.002170 |

An even safer strategy is to just bet both sides of the event, either in one account, or by having a "friend" do the same in another account. In this case, you are guaranteed to make the expected payoff, and get your money back. The casino's odds ensure they will take some money from you (about 5%, see below), but in theory, this is totally safe: bet the $6300, get back $5985 with zero risk, for a profit of $2985! Unfortunately, this is explicitly forbidden by the terms of the promotion, which means the casino can void your winnings. The promotion terms state:

19.Bonus Abuse / Fraud: Participants who engage in fraudulent activity, bonus abuse or inappropriate play, can have their Bonus Cash voided, winnings voided or withheld, be excluded from future bonuses or awards, and/or have their Caesars Account locked at the discretion of Caesars. These activities include, but are not limited to:

a.Wagering on both sides of a line on an individual game,

b.Multiple/grouped accounts linked by device/static IP/cashier/location and betting exploitatively.

For example, one game right now has odds of -260 / +210, which corresponds to 72.2% / 32.3%, which totals to 104.5%. This means the implied probability for these outcomes are both a bit too high, and therefore the casino is paying out a bit less than they should. If we adjust the probabilites to make them total 100%, they are 69.09% / 30.91%, and the expected payoff of putting $100 on both sides of the bet becomes $191.48, or a loss of 4.26%.