Using the Pythagorean Theorem to Beat the Books

If you’re like most of us the last time you heard the “Pythagorean Theorem” was high school geometry. But this 2,500-year-old formula, created by Pythagoras of Samos, to describe triangles, is a simple but effective tool for sharp sports bettors across the globe.

  • Stated plainly, the Pythagorean theorem is a^2+b^2=c^2. 

Bill James, the famed Sabermetrician, is the first person to use the theorem for sports. He applied Pythagoras’ formula to baseball in the early 1980s. James’ formulation, the “Pythagorean Expectation,” drew inspiration from Pythagoras.

The PE predicts a team’s win percentage from runs scored and allowed. It was not long before football and basketball analysts had their own version of PE.

Win Ratio=〖"scored" 〗^n/(〖"scored" 〗^n+〖"allowed" 〗^n )

The same formula applies across all sports, only the exponent (i.e. the power we raise to) changes.

How is this Formula Useful for Sports Betting?

Any serious gambler knows a good Moneyball stat doesn’t beat the line. So how do sharp bettors’ profitably use the PE? Let’s walk through an example using the Pythagorean Expectation to predict NFL season win totals.

A couple of warnings when using this method:

  1. A decade ago you could pick off a few weak season wins lines using nothing but PE. Nowadays sites like ESPN and Fox give PE numbers for each team. When a piece of information goes mainstream it loses its value. That means we need to make some tweaks to the calculation if we want to win.
  2. We’re not taking any personal changes into account. Major trades, injuries, and retirements have a large impact on season wins. When betting real money always use all available information.

Practical Example: Predicting NFL Season Win Totals

Using the Pythagorean Expectation, and a couple adjustments, we’re going to predict the Jets and Giants 2020 season win totals from their 2019 performance.

New York Giants

The Giants were dreadful in 2019. They finished the regular season 4-12.

To work out the Pythagorean Expectation we need a team’s points scored (PF) and a team’s points given up (PA). You can use season totals or averages. We’ll use totals.

New York scored 341 points in 2019 and gave up 451 points. Using those numbers we can come up with a prediction for next season’s win percentage. Let’s show our work. We need to raise both PF and PA to the power of 2.37 (that’s standard for NFL).

Filling in our formula, we can apply the numbers to find out the predicted win ratio:

  • Win Ratio=341^2.37 / (341^2.37+451^2.37 )

After calculation, we are left with a 34% estimated win percentage for the season or 5.44 games (34% x 16 games). Depending on your preference you can round up to 5.5 or down to 5.

New York Jets

The Jets 2019 season was subpar they went 7-9, only scoring 276 points while giving up 359 points. Doing the same calculation we did above we expect the Jets to win 35% of their games.

  • Win Ratio=276^2.37 / (276^2.37+359^2.37 )

If that sounds off to you it should. The Giants were 4-12 and according to their PE, we expect them to win 5.5 games. The Jets won 3 more games (a huge difference in a season of only 16 games), yet we expect them to win 5.5 games.

Either the Giants under-performed while the Jets over-performed or a little bit of both. The Pythagorean Expectation is a great tool for finding teams that were lucky and overachieving, or unlucky and catching bad breaks.

You can use this information to your advantage. Often over-performing teams end the NFL season with a good record but get knocked out of the playoffs in the first round by a lower-seeded team. The inverse happens too. Use the Pythagorean Expectation to identify those situations.

Adjusting Our Model: Adding in Turnovers

Turnovers are a huge factor in the NFL. An extra turnover in a game has a bigger impact than a home-field advantage.

Teams that turn the ball over more often lose more games. Last season the Giants averaged a -1.1 turnover margin per game (31st out of 32 teams).

The Jets, while in the bottom half of the league, average almost a whole turnover less per game, -0.2. That difference is a major reason for the win discrepancy. Let’s adjust our PE prediction by accounting for turnover rates.

We’re going to make some general assumptions:

  1. 50% of turnovers are due to luck
  2. Turnover is worth 4 points.

In 2019, the turnover margin for each team was Giants -17 and Jets -4. We’re going to add the points “lost” via turnovers to both teams. Because 50% of turnovers are due to luck divide the extra points for each team by half.

  1. Giants 17 x 4 = 68/2 = 34 points
  2. Jets 4 x 4 = 16/2 = 8 points

The Final Tweak and Results: Lets Beat the Sportsbooks

We’re going to split the turnover points adjustment in two and add half to the PF and subtract half from the PA. The reasoning is simple. Turnovers result in fewer scores but it also puts the ball in the hands of opposing offenses.

Those extra possessions turn into scores a percentage of the time. Let’s walk through the calculation.

New York Giants

  • Divide 34 points by 2.
  • Add 17 points to PF going from 341 to 358
  • Subtract 17 points from PA going from 451 to 434.
  • Recalculate the Pythagorean Expectation using the new totals.
  • The Giants go from a win expectation of 34% to 39%, 6.2 wins for the season.

New York Jets

  • Divide 8 points by 2.
  • Add 4 points to their PF going from 276 to 280
  • Subtract 17 points from the PA going from 359 to 355.
  • Recalculate the Pythagorean Expectation using the new totals.
  • The Jets go from a win expectation of 35% to 36%, 5.8 wins for the season.

Making our Bets

It’s time to go line shopping. DraftKings and SugarHouse posted season win totals for 2020-21. The odds for the Jets and Giants are in the table below.

Sportsbook

Team

Wins

Over

Under

DraftKings

NYG

6

-110

-110

SugarHouse

NYJ

6.5

-110

-110

Before we can compare our estimate to DraftKings or SugarHouse we must convert the odds into win percentages. Luckily -110 is easy to convert. It’s a 50/50 bet with some juice added in.

Using our Pythagorean win predictions and a simple binomial simulation we’re ready for the final analysis.

New York Giants

Wins

PctOver

Push

6

43%

20%

  • Based on this result betting over six wins has thin value.

New York Jets

Wins

PctOver

6.5

35% 

  • Based on this result betting under 6.5 wins is a strong play.

Summary

Remember you can’t apply the formula blindly. To find an edge we tweaked our numbers using turnover margins. But that’s not the only option.

In fact, it’s a well-known adjustment. You need to come up with your own methods to win long term.

You won’t win betting widely available information.

Using simple math we identified two potential bets. That’s only two out of 32 NFL teams. The Pythagorean Expectation works for every sport and you can apply it in your own creative ways.