# What should I do with that free bet?

Every once in a while the Sportsbook Fairy will float down from cyber-Elysium and magic a free bet into your account. How should you best take advantage of such a fortuitous circumstance? Sure you could be a well behaved, risk averse bettor and bet that money on your favorite side or total, or that -300 favorite, but that won’t maximize your expected return. I’m going to give you permission to let loose your inner degenerate, that gambler within dying to bet on longshot parlays, in the name of higher expected return.

First, I will define the problem. Sportsbooks, as a reward for long time users or a promotion for new sign-ups, will sometimes gift a free bet. Typically, the free bet is for wagering purposes only; you cannot withdraw the money. If you win the bet, you can keep the winnings only (usually subject to rollover requirements) and the free bet disappears. If you lose the bet, that’s it. The free bet disappears from your account, but you don’t lose any real money. What is the best way to maximize your expected return from this situation?

If you are an educated and disciplined sports bettor, sides and totals likely make up the majority of your bets. For good reason, these are the bets with the lowest vigorish, juice, cut, rake, take, commission, house edge, whatever you want to call it. You likely shy away from longshot parlays with large payoffs for similar reasons. These bets tend to have higher vig than straight bets. Some illustrative math follows.

For a bettor with a 50% win rate, a standard 2-team parlay paying 2.64 to 1 (2.64 is a typical payoff for offshore sportsbooks, 2.6 is typical in Las Vegas) has 9% vig:

$$ {[(.25)×(2.64×$100)]-[.75×$100]= -$9.00 } $$

25% of the time the bettor will win \$264; 75% of the time he will lose \$100, for an expected return of -$9, or -9%.

If the bettor was to make the equivalent wager using straight bets, he would lay \$100 to win \$90.91, then bet the entire \$190.91 to win \$173.55 (at -110 odds). For a bettor with a 50% win rate, the expected return on two progressive bets at -110 odds is:

$$ {[(.25)×($90.91+173.55)]- [.75×$100]= -$8.885 } $$

25% of the time the bettor will win \$90.91 on the first bet and \$173.55 on the second. The remaining 75%, he will lose \$100. This is $\frac {\$8.885}{\$100} = 8.885\%$ of the original stake, i.e. 8.885% vig.

So, the sportsbook has a higher edge on a typical bettor when he bets a 2-team parlay instead of 2 progressive sides. Please see Table 1 below for a breakdown of the house edge on parlays involving a larger number of sides.

**Table 1: Typical House Edge vs Parlay Bettor with 50% win rate (offshore sportsbooks)**

# of Teams | Payoff (to 1) | Prob Win | Parlay House Edge | Straight Bet House Edge | Difference |

2 | 2.64 | 0.2500 | 9.0% | 8.9% | 0.115% |

3 | 6 | 0.1250 | 12.5% | 13.0% | -0.526% |

4 | 12.28 | 0.0625 | 17.0% | 17.0% | 0.021% |

5 | 24.35 | 0.0313 | 20.8% | 20.8% | 0.028% |

6 | 47.41 | 0.0156 | 24.4% | 24.4% | 0.004% |

7 | 91.42 | 0.0078 | 27.9% | 27.8% | 0.003% |

8 | 175 | 0.0039 | 31.1% | 31.1% | 0.174% |

Now, that free bet changes things. First off we can’t make progressive straight bets. We are only allowed a single bet. The expected return on a single straight bet of \$100 is $\$90.91 \times .5 = \$45.455 $. Half the time, we win \$90.91. The other half, we lose nothing.It is notable that the house edge is lower on a 3-team parlay than on 3 progressive straight bets. There are ways to exploit this which is a good topic for a future article.

If we want to be more certain to win something from the free bet, we could bet on a large favorite. A favorite of -400 has an implied win probability of 80%[i] and we stand to win \$25. The expected return is $(.8) \times \$25 = \$20$, a worse return than the straight bet. If we decide to take on some extra risk for some extra return, we could bet the underdog in that same contest. Typically, an underdog in the same game as a -400 favorite would pay odds close to +325, depending on the sportsbook. The implied win probability would be 20%, 1 minus the 80% implied win probability of the favorite[ii]. The expected return would then be $(.2)×\$325=\$65$. Not bad, but we can improve that expected return by betting parlays[iii].

For a 2-team parlay, the expected return is $.25×\$264=\$66$. One quarter of the time, we win \$264 and the remaining 75%, we lose nothing. We can increase the expected return by adding more legs to the parlay as shown in Table 2.

**Table 2: Expected Return from a $100 Free Bet using Parlays**

# of Teams | Prob Win | Payoff | Expected Return |

2 | 0.2500 | $264.00 | $66.00 |

3 | 0.1250 | $600.00 | $75.00 |

4 | 0.0625 | $1,228.00 | $76.75 |

5 | 0.0313 | $2,435.00 | $76.09 |

6 | 0.0156 | $4,741.00 | $74.08 |

7 | 0.0078 | $9,142.00 | $71.42 |

8 | 0.0039 | $17,500.00 | $68.36 |

The highest expected return of \$76.75 comes from a 4-team parlay. Just keep in mind that we won’t cash that ticket very often, but when it hits; it’s yoooge. Parlays work in this case because we can leverage the free bet. In effect we can bet the free money 2, 3, 4, or more times. Our winnings compound and we don’t have to worry about losing real dollars.

Next time a free bet magically appears in your account I hope you will consider betting parlays. You get a chance to depart from being cold, disciplined, and risk averse. You can let your inner gambler out and maximize your expected return at the same time. Thank you Sportsbook Fairy!

[i] Implied win probability is the win probability needed to break even on the bet. The formula for implied win probability from a money line is:

$$Win \,Probability = \frac {amount \,risked}{( amount \, risked + amount \,won)}$$

In the example in the text :

$$\frac{400}{(400+100)} = 80%$$

The sportsbook is implying that they think there is an 80% of the favorite winning.

[ii] The implied win probability should be taken from the money line of the favorite, not the underdog. Sportsbooks short the payout on the underdog, so the odds are off. If you calculate the implied win percentage for the underdog in my example it would be $\frac{100}{425}=23.53\%$. Adding that to the win probability for the favorite equals 103.53%, over 100% probability. The book is taking a 3.53% vig in this example. A full discussion of this topic is beyond the scope of this article.

[iii] To try to improve expected return, you can experiment with larger and smaller underdogs with lesser and greater implied win percentages, respectively. I have never found any that will match the return of parlays.

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