By Kelly Galarneau

Don’t Attack Me Until the End.  My probability model is not perfect, and never will be.  We will discuss the assumptions and limitations of this model.  History and Concept  NCAA tournament  Rating Systems

Jeff Sagarin’s Ratings  Published in the USA Today since  NCAA men’s basketball tournament selections  Bowl Championship Series (BCS) selections  Secret Formula  Each team’s rating  Home field advantage (constant for all teams)

Any Given Sunday  We know that a team’s performance is not always constant. On some days they play better, others worse.  Many things naturally vary according to the normal distribution, so perhaps we can assume NFL ratings do the same.  In an , Sagarin suggested that I use a standard deviation of 15 or 16 (perhaps 15.5). I chose 16.

Normal Distribution  Here is a graph for the theoretical performance of the Tennessee Titans, with a mean of 32.7 and standard deviation of 16.

Normal Curves  Now compare two teams, Titans with average of 32.7 and the Lions with an average of We’ll pretend the Lions are playing at home and add in Sagarin’s prescribed 3.02.

Monte Carlo Simulation  Used for simulating probability situations.  Mostly used for business and finance applications.  Allows us to vary a parameter according to whatever distribution we choose (uniform, normal, Poisson, exponential, etc.)  As the parameter is changing, we can observe the effect on other variables.  I am using free software from (Rutgers University)

How each trial works  We let the computer pick a random “performance rating” from the normal distribution for each team.  We see which one is greater and tally that as a win for that team.  Then repeat 1000 times. This graph shows the results of one trial, with a win going to the Titans.

The graph on the left represents a game in which the Lions perform better than average, the Titans perform worse than average, but the Titans still get the win. The graph on the right represents a game in which the Lions pull off the upset. Then we repeat… …1000 times.

Here is a graph of 1000 trials

Here is my actual Excel programming: ABCDEFG 1 TeamRatingS.D. NormalIf/thenOutput 2 Titans =gennormal(C2,D2)=if(E2>E3,1,0)=simoutput(F2) 3 Lions =gennormal(C3,D3)=if(E3>E2,1,0)=simoutput(F3) ABCDEFG 1 TeamRatingS.D. NormalIf/thenOutput 2 Titans Lions Here is one trial: This matchup gives us: Titans 88.8% (1000 trials) Lions 11.2%

Do we need a simulation?  I tried to approach the probability calculation from a theoretical perspective.  My thought was to assign each team a random variable X and Y and vary them according to a normal distribution. So: X ~ N(μ₁, σ) and Y ~ N(μ₂, σ).  A statistics textbook led me to the idea of multiplying them to get a 3-D probability distribution [X, Y, and P(X,Y)].

A 3-D Probability Distribution Here, X ~ N(35.72,16) and Y ~ N(10.84,16) Where is X > Y? YX Titans 88.8% Lions 11.2%

Integrals  To find the area under this surface, we need to evaluate the double integral.  Mathematica will not evaluate the exact integral, but it will give us a decimal approximation.

Comparing the results  Mathematica Input : NIntegrate[NIntegrate[(1/((16^2)*2*Pi))*(E^((- 1/2)*((x-35.72)/16)^2))*(E^((-1/2)*((y- 7.82)/16)^2)),{y,x,300}],{x,-300,300}]  Mathematica gives us: Titans 89.1% Lions 10.9%  The Monte Carlo with 1000 trials gave us: Titans 88.8% Lions 11.2%

The Bracket  6 teams from each conference (AFC and NFC) make the playoffs. The division winners are seeded 1-4 by record, the wild card teams are seeded 5 and 6 by record.  In each game the higher seed gets home field advantage.  In the “Wild Card” round of the playoffs:  The 3 seed plays the 6 seed  The 4 seed plays the 5 seed.  The 1 and 2 seeds get a first round bye.  In the “Divisional” round the 1 seed gets to play the lower seed of the two advancing teams.

So that means… or seed wins 6 seed wins

Possibilities…  In a playoff bracket with 12 teams, there are 11 games to be played, with a total of 2^11=2048 possibilities.  In my Excel spreadsheet I have made use of the complement rule and referencing so that I only need to calculate 66 matchups.

Assumptions and Limitations  We are using offensive and defensive statistics from the regular season, we are assuming that teams continue similar play into the post-season.  We are assuming that overall performance is normally distributed.  We are assuming that home field advantage is the same constant for all teams.  We are not taking anything into account for weather, injuries, matchups that might be significant, or any other variables.

2007 trial run (AFC) SeedTeamRating Div. Game Champ. Superbowl Superbowl Win 1pats % 73.9% 48.3% 32.4% 2colts % 58.0% 25.1% 13.7% 3chargers % 28.6% 11.9% 6.5% 4steelers % 12.8% 4.3% 1.9% 5jags % 19.2% 8.0% 4.0% 6titans % 7.4% 2.5% 1.0% %

2007 trial run (NFC) SeedTeam Rating Div. Game Champ. Superbowl Superbowl Win 1cowboys % 62.6% 34.9% 14.5% 2packers % 65.9% 35.6% 15.6% 3seahawks % 18.4% 7.3% 2.3% 4bucs % 12.0% 4.2% 1.2% 5giants % 26.4% 12.7% 5.1% 6redskins % 14.7% 5.2% 1.7% %

2008 Playoff Prediction  My OFFICIAL prediction won’t be available until after week 17 (end of regular season is Sunday, Dec. 28)  Over Christmas break you can access it at: kgalarneau.wikispaces.com/NFL+Prediction kgalarneau.wikispaces.com/NFL+Prediction  Feel free to me with your comments or suggestions:

2008 Prediction (As of Dec. 15) SeedTeamRating Div. Game Champ. Superbowl Superbowl Win NFC1Giants % 63.6% 37.7% 19.0% 2Panthers % 57.5% 28.1% 13.5% 3Vikings % 25.4% 11.8% 5.3% 4Cardinals % 16.3% 5.8% 2.2% 5Cowboys % 20.0% 8.4% 3.6% 6Bucs % 17.2% 8.2% 3.7% sum % AFC1Titans % 64.6% 39.6% 22.0% 2Steelers % 65.1% 31.8% 16.9% 3Jets % 14.7% 5.1% 2.0% 4Broncos % 9.7% 2.8% 1.0% 5Colts % 24.3% 10.8% 5.4% 6Ravens % 21.5% 9.9% 5.3% sum % standard dev. = 16 home field advantage = 2.58

QQuestions? Comments?

But Wait… …There’s More!

History of Seeds  The NFL has been using the 12 team playoff system since SeedDivisionChamp.SuperbowlSuperbowl Win sum

By Percentage: SeedDivisionalChampSuperbowlSuperbowl Win 138.9%50.0%44.4% 237.5%27.8% 369.4%9.7%5.6% 466.7%6.9%11.1% 533.3%5.6%2.8%5.6% 630.6%1.4%2.8%5.6% sum2111

Superbowl Winners by Seed since 1990 First 10 years – 7 #1 seedsLast 8 years – Only 1 #1 seed 9/10 are 1 or 2 seeds4/8 are 1 or 2 seeds

The End. Special thanks to: Harry Geiser Steve Havlichek