King Yao chimes in with his plan to buy Patriots futures and then bet against them in individual games, planning to arbitrage the two markets for a guaranteed profit.

I found a similar strategy useful during the World Series, although my reasoning was that the futures odds were off, not the individual game odds. As it turned out, betting the Red Sox to win the series and the Rockies in each game would have yielded a tidy 5% profit for a week's work.

Given that the Pats are reportedly 1-8 favorites to win the Super Bowl at The Mirage, it looks like King will need to use a betting exchange to find a good price on them to win it all. If only Vegas let you bet the "no" on those ridiculous lines...

## Monday, December 31, 2007

### Updated NFL Probabilities

Team | Conf | SB |

NE | 65.4 | 54.4 |

Ind | 22.2 | 15.9 |

SD | 5.7 | 3.7 |

Pit | 2.4 | 1.5 |

Jax | 3.7 | 2.5 |

Ten | 0.6 | 0.2 |

Dal | 57.8 | 13.4 |

GB | 24.3 | 4.9 |

Sea | 7.7 | 1.5 |

TB | 4.3 | 0.7 |

NYG | 3.6 | 0.7 |

Was | 2.3 | 0.4 |

I think if there's one general lesson about playoff futures betting, it's that you don't want to bother betting on a 6 seed to win the Super Bowl; the Redskins are going off around 50-1 when they should be 250-1. By betting the Skins game-by-game, you'll get a much better price than 50-1.

Actually, maybe another lesson is not to bet an NFC team to win the Super Bowl this year; only the Cowboys have a payoff that's close to fair odds.

(edited to fix rounding errors)

## Monday, December 24, 2007

### What the Patriots Can Teach Us About Futures Odds

A coin has landed on heads 15 times in a row. What are the chances of it landing heads on the 16th flip?

Many of you will pick one of three answers:

- The coin is "hot", so the chances are greater than 50%.

- The coin is fair, so the chances are exactly 50%.

- The coin is "due", so the chances are less than 50%.

These answers are all using incorrect logic. The correct answer considers the possibility of a loaded coin. Loaded coins are rare enough that when we start flipping, we don't think about them. But when 15 consecutive heads are flipped, Bayesian probability tells us that the chances of a loaded coin have gone way up. Thus, the probability of a heads on the 16th flip is significantly greater than 50%.

Now think about analyzing the probability of a team going 16-0 before the season starts. If you're a good handicapper, you can estimate the moneyline for each game and multiply the team's winning probabilities together. But this wouldn't give you an accurate estimate.

If a team is 15-0, chances are very good that the team is much better than you believed at the start of the season. Certainly this is the case for the 2007 Patriots; their season wins over/under was set at 11.5. Thus, your preseason estimate for the 16th game is likely to underestimate their chances of winning. The same is true for other intermediate games. So if you really want to handicap the chances of an undefeated team, or a similar proposition, you must account for the probabilities of teams performing much better or worse than you expected.

This principle has other applications as well. Many preseason baseball simulations have 80-projected-wins teams almost never making the playoffs (except in the NL Central, of course). But 80 is just a mean estimate; there's a significant chance the team actually has 88-win talent. A team that has a fixed talent level of 80 wins will make the playoffs very rarely, but if their actual talent follows a normal distribution from 70 to 90 wins, they have a legitimate shot. (Remember, a 90-win team can still overperform and win 100 games.)

And that, kids, is how the Colorado Rockies made it to the World Series. Colorado probably won't repeat, but I'd bet that at least one ".500 team" will play in October next year.

Many of you will pick one of three answers:

- The coin is "hot", so the chances are greater than 50%.

- The coin is fair, so the chances are exactly 50%.

- The coin is "due", so the chances are less than 50%.

These answers are all using incorrect logic. The correct answer considers the possibility of a loaded coin. Loaded coins are rare enough that when we start flipping, we don't think about them. But when 15 consecutive heads are flipped, Bayesian probability tells us that the chances of a loaded coin have gone way up. Thus, the probability of a heads on the 16th flip is significantly greater than 50%.

Now think about analyzing the probability of a team going 16-0 before the season starts. If you're a good handicapper, you can estimate the moneyline for each game and multiply the team's winning probabilities together. But this wouldn't give you an accurate estimate.

If a team is 15-0, chances are very good that the team is much better than you believed at the start of the season. Certainly this is the case for the 2007 Patriots; their season wins over/under was set at 11.5. Thus, your preseason estimate for the 16th game is likely to underestimate their chances of winning. The same is true for other intermediate games. So if you really want to handicap the chances of an undefeated team, or a similar proposition, you must account for the probabilities of teams performing much better or worse than you expected.

This principle has other applications as well. Many preseason baseball simulations have 80-projected-wins teams almost never making the playoffs (except in the NL Central, of course). But 80 is just a mean estimate; there's a significant chance the team actually has 88-win talent. A team that has a fixed talent level of 80 wins will make the playoffs very rarely, but if their actual talent follows a normal distribution from 70 to 90 wins, they have a legitimate shot. (Remember, a 90-win team can still overperform and win 100 games.)

And that, kids, is how the Colorado Rockies made it to the World Series. Colorado probably won't repeat, but I'd bet that at least one ".500 team" will play in October next year.

## Sunday, December 23, 2007

### Why You Can't Always Trust CoolStandings

Check out their current NFL odds.

There's nothing fundamentally wrong with their algorithm; it just isn't considering motivation. The Redskins and Titans each play a superior team in Week 17, but their opponents, the Cowboys and Colts, will be resting their starters for the playoffs while Washington and Tennessee play their hearts out to get their respective 6 seeds.

Tennessee opened as a 4-point favorite against Indy, and would probably be about a 7-point dog if Indy was trying. That's a huge swing, enough to make them substantial favorites instead of severe underdogs. Washington would ordinarily be an underdog to Dallas, but facing an unmotivated Cowboys team at home, the Redskins will probably open as solid faves.

Here are my updated odds estimates:

Edit: If you're wondering how the Giants' chances managed to get worse despite their clinching a playoff spot, bear in mind that they were almost certain to get in the playoffs anyway, plus I tweaked the numbers some.

That Titans number is not a typo. To win the AFC, they have to win at Indy, at San Diego (or Pittsburgh), at New England, and probably at Indy again. That's one brutal path.

There's nothing fundamentally wrong with their algorithm; it just isn't considering motivation. The Redskins and Titans each play a superior team in Week 17, but their opponents, the Cowboys and Colts, will be resting their starters for the playoffs while Washington and Tennessee play their hearts out to get their respective 6 seeds.

Tennessee opened as a 4-point favorite against Indy, and would probably be about a 7-point dog if Indy was trying. That's a huge swing, enough to make them substantial favorites instead of severe underdogs. Washington would ordinarily be an underdog to Dallas, but facing an unmotivated Cowboys team at home, the Redskins will probably open as solid faves.

Here are my updated odds estimates:

Team | Conf | SB |

NE | 67.8 | 54.6 |

Ind | 21.2 | 15.1 |

Pit | 4.1 | 2.6 |

SD | 4.3 | 2.6 |

Jax | 2.3 | 1.4 |

Ten | 0.4 | 0.1 |

Dal | 56.6 | 15.0 |

GB | 25.2 | 5.4 |

Sea | 6.6 | 1.2 |

TB | 4.5 | 0.7 |

NYG | 3.7 | 0.7 |

Was | 2.4 | 0.4 |

Edit: If you're wondering how the Giants' chances managed to get worse despite their clinching a playoff spot, bear in mind that they were almost certain to get in the playoffs anyway, plus I tweaked the numbers some.

That Titans number is not a typo. To win the AFC, they have to win at Indy, at San Diego (or Pittsburgh), at New England, and probably at Indy again. That's one brutal path.

## Wednesday, December 19, 2007

### Tom Brady Redux

Tom Brady now needs five touchdown passes in the final two weeks to break Peyton Manning's record of 49. Will he get there?

The bettors at World Sports Exchange certainly think so; they've made Brady more than a 2-1 favorite to get to 50. But the betting market for obscure props like this one is rarely efficient; the ratio of 'fan money' to sharp bets is unusually high. So we can't trust the market for the answer to this one.

The big unanswered question, of course, is how much action Brady will see in Week 17 if the Patriots are still undefeated at that point. The Giants are heavily favored to clinch a playoff spot this weekend, so they likely won't be trying their hardest to stop the Pats' juggernaut. Will that convince Bill Belichick to coast to a seventeenth win by letting his backups play the second half? Will the Pats play it safe either way and not risk an injury to Brady, the one player they truly can't afford to lose?

I'll operate under these assumptions:

- If the Pats lose to the Dolphins (roughly a 6% chance), Brady sits almost all of Week 17.

- If the Pats are 15-0 and the Giants have clinched a playoff berth (66%), Brady plays with this frequency:

1 quarter: 20%

1st half: 10%

3 quarters: 15%

full game: 55%

- If the Pats are 15-0 and the Giants are still playing to get in (28%), Brady plays with this frequency:

1 quarter: 15%

1st half: 10%

3 quarters: 5%

full game: 70%

As for Brady's true skill level of throwing touchdowns, I'm going to set it at 2.5 per game, or 40 per season. I think this is fair. Yes, he's well over that pace this season, but his previous career high was 28, and only three other QBs in history have thrown 40 over a full season. Regression to the mean is a powerful force.

I'll spare you the boring calculations, but the Poisson distribution tells us that with this set of assumptions, Brady breaks the record 44.2% of the time, a far cry from 2/3.

If Brady is assumed to play all eight quarters of his remaining games, he breaks the record with 56.0% frequency.

Using a true rate of 50 TD/season and the above playing time distribution, Brady breaks the record 62.0% of the time.

If you assume eight quarters of play and a true rate of 50 TD/season, that number increases to 74.7%, but I think that's unreasonable on both counts.

The bettors at World Sports Exchange certainly think so; they've made Brady more than a 2-1 favorite to get to 50. But the betting market for obscure props like this one is rarely efficient; the ratio of 'fan money' to sharp bets is unusually high. So we can't trust the market for the answer to this one.

The big unanswered question, of course, is how much action Brady will see in Week 17 if the Patriots are still undefeated at that point. The Giants are heavily favored to clinch a playoff spot this weekend, so they likely won't be trying their hardest to stop the Pats' juggernaut. Will that convince Bill Belichick to coast to a seventeenth win by letting his backups play the second half? Will the Pats play it safe either way and not risk an injury to Brady, the one player they truly can't afford to lose?

I'll operate under these assumptions:

- If the Pats lose to the Dolphins (roughly a 6% chance), Brady sits almost all of Week 17.

- If the Pats are 15-0 and the Giants have clinched a playoff berth (66%), Brady plays with this frequency:

1 quarter: 20%

1st half: 10%

3 quarters: 15%

full game: 55%

- If the Pats are 15-0 and the Giants are still playing to get in (28%), Brady plays with this frequency:

1 quarter: 15%

1st half: 10%

3 quarters: 5%

full game: 70%

As for Brady's true skill level of throwing touchdowns, I'm going to set it at 2.5 per game, or 40 per season. I think this is fair. Yes, he's well over that pace this season, but his previous career high was 28, and only three other QBs in history have thrown 40 over a full season. Regression to the mean is a powerful force.

I'll spare you the boring calculations, but the Poisson distribution tells us that with this set of assumptions, Brady breaks the record 44.2% of the time, a far cry from 2/3.

If Brady is assumed to play all eight quarters of his remaining games, he breaks the record with 56.0% frequency.

Using a true rate of 50 TD/season and the above playing time distribution, Brady breaks the record 62.0% of the time.

If you assume eight quarters of play and a true rate of 50 TD/season, that number increases to 74.7%, but I think that's unreasonable on both counts.

## Sunday, December 16, 2007

### Updated NFL Probabilities

Standard disclaimers apply.

Team | Conf | SB |

NE | 67.4 | 54.1 |

Ind | 21.7 | 15.0 |

Pit | 4.2 | 2.7 |

SD | 3.8 | 2.2 |

Jax | 2.1 | 1.2 |

Dal | 56.3 | 15.3 |

GB | 26.1 | 5.7 |

Sea | 6.7 | 1.2 |

TB | 4.0 | 0.7 |

NYG | 4.4 | 0.8 |

## Friday, December 14, 2007

### What's On Tap For 2008

- I'm moving off Blogger and onto "real" hosting. If you have the Blogger address bookmarked, please adjust it to www.baseballplayoffodds.com, which currently redirects to this page.

- Right now, I'm working with someone on developing a Monte Carlo simulation like the ones you see on BP and Coolstandings, only better. I can't promise anything about the functionality yet, but the inputs will definitely be more accurate, since I can better account for injuries, trades, and updated player/team forecasts.

- For the playoffs and late in the regular season, the Monte Carlo simulation can give way to a mathematical analysis based on estimated odds for each individual game. Among other things, this should eliminate the problem of variance in quality of starting pitchers. (If your team is starting Johan Santana and Jose Lima in its two remaining games, they should be assigned winning chances of 65% and 35%, not 50% for each game.)

If nothing else, it should be interesting to look at.

- Right now, I'm working with someone on developing a Monte Carlo simulation like the ones you see on BP and Coolstandings, only better. I can't promise anything about the functionality yet, but the inputs will definitely be more accurate, since I can better account for injuries, trades, and updated player/team forecasts.

- For the playoffs and late in the regular season, the Monte Carlo simulation can give way to a mathematical analysis based on estimated odds for each individual game. Among other things, this should eliminate the problem of variance in quality of starting pitchers. (If your team is starting Johan Santana and Jose Lima in its two remaining games, they should be assigned winning chances of 65% and 35%, not 50% for each game.)

If nothing else, it should be interesting to look at.

## Wednesday, December 12, 2007

### Updated Projections

My predicted 2008 standings generated some criticism, fair and otherwise, but no one mentioned one error I'm going to correct here: I had the average AL team winning 80 games and the average NL team 82.

Each team plays a maximum of 18 interleague games, so these numbers shouldn't stray far from 81, but clearly the AL is the much stronger league at this point, so I re-ran the numbers with the AL averaging 81.8 wins and the NL 80.3, plus some number tweaking and roster updates through the signing of Fukudome:

Interesting how the 2008 White Sox project stronger than the 2007 squad even though they're coming off a season with 18 fewer wins. Some possible theories on why:

- Bounce-back seasons by Mark Buehrle and Javier Vazquez

- Ditching Scott Podsednik and Darin Erstad (and adding Carlos Quentin)

- Inaccurate inputs by me

Rumor has it that Fukudome is considering wearing a jersey that reads "Kosuke" much like Ichiro. I wonder how much that will hurt merchandise sales?

Each team plays a maximum of 18 interleague games, so these numbers shouldn't stray far from 81, but clearly the AL is the much stronger league at this point, so I re-ran the numbers with the AL averaging 81.8 wins and the NL 80.3, plus some number tweaking and roster updates through the signing of Fukudome:

NYY | 92 | Cle | 88 | LAA | 84 |

Bos | 91 | Det | 86 | Oak | 79 |

Tam | 84 | Min | 84 | Sea | 77 |

Tor | 84 | ChW | 76 | Tex | 72 |

Bal | 73 | KC | 72 | ||

NYM | 90 | ChC | 85 | SD | 87 |

Phi | 87 | Mil | 83 | LAD | 84 |

Atl | 83 | Cin | 82 | Ari | 79 |

Fla | 74 | StL | 75 | Col | 79 |

Was | 74 | Hou | 76 | SF | 74 |

Pit | 74 |

Interesting how the 2008 White Sox project stronger than the 2007 squad even though they're coming off a season with 18 fewer wins. Some possible theories on why:

- Bounce-back seasons by Mark Buehrle and Javier Vazquez

- Ditching Scott Podsednik and Darin Erstad (and adding Carlos Quentin)

- Inaccurate inputs by me

Rumor has it that Fukudome is considering wearing a jersey that reads "Kosuke" much like Ichiro. I wonder how much that will hurt merchandise sales?

## Sunday, December 9, 2007

### Conference/Super Bowl Odds

These are based on my estimated moneylines for playoff games, so they reflect any built-in market bias, plus my estimates may be inaccurate. Translation: Experimental, use at your own risk.

Note that the odds will not sum to 100% because I ignored teams with small chances to advance.

Note that the odds will not sum to 100% because I ignored teams with small chances to advance.

Team | Conf | SB |

NE | 69.0 | 55.0 |

Ind | 18.8 | 12.8 |

Pit | 6.9 | 4.5 |

SD | 2.4 | 1.4 |

Jax | 1.9 | 1.1 |

Dal | 58.9 | 16.3 |

GB | 24.0 | 5.1 |

Sea | 6.8 | 1.2 |

TB | 4.6 | 0.7 |

NYG | 4.1 | 0.7 |

## Thursday, December 6, 2007

### Extremely Preliminary Projections

Here's a teaser of my projected 2008 MLB standings. Keep in mind: these are very raw, as there's still plenty of transactions left between now and Opening Day, plus the numbers need fine-tuning from better projection systems, which aren't out yet:

I'm sure a lot of you are surprised at the high win totals for the Rays, Twins, Reds, and Padres, and the low totals for the Tigers and Angels. Here's a brief rundown on each:

Rays: Should get a huge boost from improved team defense this year, plus they have lots of good young players on the rise. They've also upgraded their pitching through several trades and the signing of Troy Percival.

Twins: Projection still includes Johan Santana and Joe Nathan. They still have a dominant pitching staff, and Mauer and Morneau give them some life on offense.

Reds: Loaded with young talent. Jay Bruce, Joey Votto, Homer Bailey, and Johnny Cueto are all top-shelf prospects who should make an impact in 2008.

Padres: Their hitters are better than you think--thanks, PETCO Park--and those pitchers are still devastating.

Tigers: I covered their 2008 prospects here. The trade with the Marlins doesn't help them as much as you may think, because Cabrera can't play defense and Willis is really no better than Miller going forward. I do think they'll win more than 83 games, but less than 90.

Angels: They only really have one solidly-above-average position player. Again, I think this number is a little low, but only by a couple of wins, not ten.

What's really impressive, at least to me, is that no team is projected to finish more than ten games away from .500. Of course, variance during the actual 2008 season will stretch these numbers out, so we'll probably see a team win or lose 100 games. Still, this is food for thought if you still believe baseball needs a salary cap to keep things competitive.

NYY | 91 | Cle | 86 | LAA | 82 |

Bos | 89 | Min | 83 | Oak | 77 |

Tam | 83 | Det | 83 | Sea | 77 |

Tor | 82 | ChW | 73 | Tex | 71 |

Bal | 73 | KC | 72 | ||

NYM | 91 | ChC | 85 | SD | 89 |

Phi | 89 | Mil | 84 | LAD | 85 |

Atl | 85 | Cin | 83 | Ari | 81 |

Fla | 76 | StL | 79 | Col | 80 |

Was | 75 | Hou | 79 | SF | 76 |

Pit | 75 |

I'm sure a lot of you are surprised at the high win totals for the Rays, Twins, Reds, and Padres, and the low totals for the Tigers and Angels. Here's a brief rundown on each:

Rays: Should get a huge boost from improved team defense this year, plus they have lots of good young players on the rise. They've also upgraded their pitching through several trades and the signing of Troy Percival.

Twins: Projection still includes Johan Santana and Joe Nathan. They still have a dominant pitching staff, and Mauer and Morneau give them some life on offense.

Reds: Loaded with young talent. Jay Bruce, Joey Votto, Homer Bailey, and Johnny Cueto are all top-shelf prospects who should make an impact in 2008.

Padres: Their hitters are better than you think--thanks, PETCO Park--and those pitchers are still devastating.

Tigers: I covered their 2008 prospects here. The trade with the Marlins doesn't help them as much as you may think, because Cabrera can't play defense and Willis is really no better than Miller going forward. I do think they'll win more than 83 games, but less than 90.

Angels: They only really have one solidly-above-average position player. Again, I think this number is a little low, but only by a couple of wins, not ten.

What's really impressive, at least to me, is that no team is projected to finish more than ten games away from .500. Of course, variance during the actual 2008 season will stretch these numbers out, so we'll probably see a team win or lose 100 games. Still, this is food for thought if you still believe baseball needs a salary cap to keep things competitive.

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