If you’ve watched a single uninterrupted TV commercial break in the past couple years you’ve probably noticed the deluge of sports betting ads. The industry is growing incredibly quickly. It has brought in well over $300 billion in wagers since the Supreme Court’s 2018 decision to effectively decriminalize it at the federal level.

The industry is fascinating but in this post I want to write about a specific bet-related math error.

1. The background.

Sports books make a profit by charging vigorish (usually called vig), which is just a euphemism for the fee they charge to place a bet. For example, if you wanted to place a $100 bet on the Chiefs to win by 7 points, you’d typically hand the book $110 upfront. If you win they’d hand you $210—your $110 stake plus a $100 profit—but if you lose then the entire $110 is theirs to keep. That extra $10 is the vig. The exact percentage varies depending on the situation but that’s generally how it works.

Some enterprising bettors might grow tired of paying vig and decide to set up bets between themselves. With -110 vig (the bet described above), you’d have to be correct 52.38% of the time to break even. But if you remove the vig that number drops to 50%. Bettors can cut out the middle man and essentially flip coins for fun.

2. The example.

Imagine a game where the odds are a little more interesting. Say the Chiefs moneyline this weekend is -160/+120. That means to bet on the Chiefs you’d pay -160 (risk $160 to win $100), and to bet against them you’d get +120 odds (risk $100 to win $120).

How would two parties set up a bet between themselves with no book involved? A lot of people would likely split the difference and call it -140/+140. Person A risks $140 and Person B risks $100. They both get better odds than dealing with a sports book so everybody wins. Right?

3. The problem.

While that’s true, the problem is that one party benefits more than the other.

Let me state the important part first: you can’t treat moneylines (-110, +150, etc.) like whole numbers and simply average them. Moneylines imply a certain win probability at which the bet is a neutral-EV breakeven proposition. As you can see in the plot below, implied win probability does not scale linearly with moneyline:

Intuitively it seems the average of -150 and -550 should be -350, but that’s not the case. The true average probability is closer to the smaller-magnitude number. In practice that means the person who takes the underdog side gets a better deal!

Remember, both sides are getting a discount compared to betting with a sports book. That’s great. The issue is that the underdog bettor gets a bigger discount.

Moneylines look like regular numbers but you can’t treat them that way. It’s similar to saying the average of 1/4 and 1/2 is 1/3. No it’s not. It’s 3/8. It only seems more obvious because you have a better grasp of how fractions work.

4. The solution.

Let’s get back to our Chiefs example where the odds are -160/+120. We can’t treat moneylines like regular numbers and average them, so let’s change them into a form we can easily work with: percentages.

The formula for converting a moneyline to an implied win probability is fairly simple. It changes slightly depending on whether the moneyline is negative or positive:

To convert -160 to a win probability, use the left equation. The vertical bars mean absolute value so change the negative sign to positive:

P  =  |-160| / (|-160| + 100)  =  160 / 260  =  0.615

To convert +120 to a win probability, use the right equation:

P  =  100 / (120 + 100)  =  100 / 220  =  0.455

Note that these numbers are probabilities ranging from 0.0 to 1.0. To convert to percentages you would multiply by 100. So our -160 moneyline becomes 61.5% and +120 becomes 45.5%.

But wait, those percentages add up to more than 100%. Both outcomes can’t happen that often. At this point we’ve identified how sports books come out ahead: overround. 61.5% and 45.5% add up to 107% and that extra 7% is called the overround. As you’d guess, a bigger overround means the book is taking a larger cut of the action, which makes it more difficult for bettors to clear a profit in the long run.

We can resolve the overround problem by dividing both numbers by 107% (or 100 plus whatever the overround happens to be):

0.615 / 1.07  =  0.575

0.455 / 1.07  =  0.425

Now the overround is removed and our win probabilities add up to 100%. We know the true chance of each event occuring—a Chiefs win or a Chiefs loss—according to the odds.

With our true implied win probabilities in hand we can convert back to moneylines and arrange the vig-free bet. The formula for this conversion is:

To convert 57.5% to moneyline, use the left equation:

M  =  0.575 / (1 - 0.575) * -100  =  -135

To convert 42.5% to moneyline, use the right equation:

M  =  (1 - 0.425) / 0.425 * 100  =  +135

The magnitude of the numbers is the same (-135/+135). That’s an indication that it’s a vig-free line, which was our goal. We started with moneyline odds of -160/+120 and found that when you remove the vig, the line becomes -135/+135. Person A can wager $135 and Person B can wager $100.

Note that these odds are slightly different than our original guess of -140/+140. The difference is small but important. Margins are extremely thin in sports betting and you’d be wise not to give away value for free.

5. The plug.

If you have an Android device you can download my free app from the Google Play store. It takes care of all these calculations. It also calculates half-point values and supports decimal odds for the non-Americans.

I don’t make any money from the app. There are no ads and it doesn’t track you. It was a personal project and I’m glad that other people are able to use it.