A Math Primer on NFL Moneyline Bets

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A Math Primer on NFL Moneyline Bets

An old cliche is that books do not have a dog in the fight because the vig ensures their profitability. That is not entirely true for several reasons. The theory states that if the book obtains an equal amount of dollars bet on each side, it will win no matter how the game turns out. That is true. If five people bet the favorite and five people bet the dog to win $100 each, the book will have collected $1,100. Only five people will win. So the books’ liability is $1050. That results in a $50 profit for the book. However, it is nearly impossible to get the equilibrium of wagers on each side of a play. Books also take sides intentionally. 

However, nowhere is the cliche less accurate, nor more easily explained, than when it comes to NFL moneyline bets. In short, books do have a dog in these fights—namely, the dog. Most contests will have a positive moneyline for one team and a negative one for the other. The negative numbered team is the favorite and represents the amount you have to wager to win $100. The positive team is the underdog and represents the amount you will win if you wager $100. This is all well understood. 

Now let’s examine a current line from this week and play out the scenario from the book’s perspective. I have chosen the following line from MGM in the game between the Cardinals and the Titans. I selected this game because the moneyline wagers are split close to evenly between the two teams. 

Arizona Cardinals +135

Tennessee Titans -165

Suppose MGM gets ten moneyline bets on each team. Each bet is either to win $100 on the favorite Titans or to risk $100 on the underdog Cardinals. Before kickoff, the book is holding $1,650 in Titans money and $1,000 in Cardinals money. The book has a total of $2,650 on the game. 

If the Titans win, they will have to pay out 10 x $265, $2,650. This amounts to everything they are holding on the game. If the Cardinals win, the book will have to pay out 10 x $235, $2,350. The book will be left with a $300 profit for their trouble. Think about the implications of this throughout a season. Every time the favorite wins, the book breaks even. Every time a dog wins throughout a season, the book makes bank. 

You cannot tail the book on this road to profit because you do not have the luxury of breaking even with the favorite wins. The question becomes, how do you know when to take the dog on the moneyline. The lion’s share of that answer will be determined by the method you employ to take a side on any wager: your capping methodology. However, there are two things you should know and immediately add to your analysis, and they both involve a little math. 

The first is simple, and I assume most people reading this article understand it well. The moneyline number is easily converted into a win percentage. One needs to know what that win percentage is, often called implied probability, before you can even consider making a wager. The math for converting the number is slightly different for a favorite and a dog. I will demonstrate both by using the game highlighted above. 

To determine the implied probability for the favored Titans, you would work the following equation:

-165/(-165-100)

-165/-265

= Titans implied probability is 62.26%

To determine the implied probability for the underdog Cardinals, the math is similar but slightly different.

100/(135 + 100)

100/235

= Cardinals implied probability is 42.55%

Now that we know the implied probability, the break-even numbers for each wager, we must compare that in a meaningful way to our methodology. The answer is not as simple as it may seem. Yes, you could take a position every time your probability is greater than the implied probability. 

If your numbers are correct and you employ this strategy over an infinite number of games, you will profit. The problem is that there are only a total of 289 regular-season games in a year. There are far fewer games where you would find a bettable advantage. This approach would is subject you to extreme variance within a single season because of the limited number of games (trials).  

For the sake of an example, let’s assume a few numbers. Suppose that you have the Cardinals at 45% to win the game. Suppose that you will bet twenty similar moneyline wagers in a season. Finally, let us agree that you will need to win nine of the 20 wagers you place in a season in order to be profitable. If you only win eight, you will lose $120. 

To determine whether or not you should make this wager, you need to utilize binomial probability. The formula for the calculation is below: 

Fortunately, you do not need to know that, as there are plenty of free binomial distribution calculators. So I took our numbers and plugged them into one. The results appear directly below.

The binomial calculator above explains what is likely to happen with this wager in a more significant number of trials, over the course of your 20 similar bets. It reveals the big picture to you. To explain briefly, the fourth line indicates the percentage of time that you will win precisely nine wagers, 17.7 percent. The fifth line provides the probability that you will win less than nine wagers, 41.3 percent. The sixth line provides the cumulative probability of winning nine or fewer bets in a season, 59.1%. The seventh line provides the likelihood that you will win more than nine games, 40.8%. The eighth line is the most important, it displays the probability that you win will nine or more games in a season, 58.5 percent. 

While a 58.5 percent winning record is successful, it is not enough of an edge to make the pursuit of such bets worthwhile when you consider its implications. The number means that over the next decade you will likely have between four and five losing seasons. You can find a better proposition for your money than what this provides. 

I need to have an eighth line “Cumulative Probability,” which is over 70 percent before I will take a side on a moneyline. Applying my standards to the current game between the Cardinals and the Titans, I would need my model to conclude that Arizona has a 49% probability of winning the game. (see the chart below). This results in a winning season 71.8% of the seasons. 

Remember that the winning percentage needed to create a cumulative probability above 70 percent will change in every matchup based on the moneyline. Although I used an underdog in this example, the math works the same with a favorite. It is a worthwhile wager as long as your numbers are accurate and the cumulative probability is above 70 percent. 

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