 With the Super Bowl between the Patriots and Falcons coming up on Sunday, the internet is flooded with probabilities of which team will win as well as a myriad of other prop bets. Currently, the Patriots are favored by three points and the predicted score is estimated to be around 28-25. Nate Silver of the FiveThirtyEight blog, using a pretty intense simulator, has the Patriots at 61% of winning the Super Bowl with the Falcons consequently at 39%. His forecast is based on 100,000 simulations of the season with updates after every game. His model uses Elo ratings, a complicated measure of strength based on head-to-head results and quality of opponents, to calculate a team’s chances of winning their next game. To check these probabilities, we developed a basic simulator incorporating only the final scores from games played by the Patriots and Falcons during the regular season and playoffs so far.
The points scored by each team represents their offense. The points scored against each team represents their defense. Fitting a probability distribution to each of these four scores allows us to randomly sample from each statistic and determine an appropriate total score for each team over as many Super Bowl simulations as we want. The two most common probability distributions for sports scores are the Poisson distribution and Negative Binomial distribution. To decide on which to use, let us examine the histograms:  The spreads are all over the place due to their relatively small sample sizes. Here are the descriptive statistics:
Since the means and variances are not the same, we can probably rule out the Poisson distribution. As another blow to the Poisson distribution, the Negative Binomial has been proven to accurately describe scores in baseball, soccer, rugby, and college football. Because of this analysis, we decided to model the scores above using the Negative Binomial distribution. The Negative Binomial requires two inputs, the size or dispersion parameter (the shape parameter of the gamma mixing distribution) and the probability of success where prob = size/(size+mean). Since we know the mean of each statistic, we will compare 1000 samples of various sizes to see which one fits the best.  The highest single game score by a team is 72 points, by Washington in 1966. Thus we can rule out sizes with scores higher than that, which is anything around a size of 4. By the looks of the graphs, both size 50 and size 100 seem to be decently accurate, at least for the Falcons Offense. It turns out the difference in the final results between the two are negligible so because of this, we will just use a size of 100 to describe all of our distributions. Now we are ready to simulate the Super Bowl!
. For each simulation of the Super Bowl, we took a single random sample from each distribution to get a number describing each of the Patriot’s offense, Patriot’s defense, Falcon’s offense, and Falcon’s defense. Then we took the mean of the Patriot’s offense score and the Falcon’s defense score to obtain a score representing the Patriot’s total points and the mean of the Falcon’s offense score and Patriot’s defense score to get a score of the Falcon’s total points. Then we compared the two, threw out ties, and recorded who won. After simulating 100,000 Super Bowls in this way, our method returned the Patriots winning 61% and the Falcons winning 39%. The average score by the Patriots was 27 points and the average score by the Falcons was 25 points. These results are on-par with Vegas odds and matches almost exactly the winning probabilities that Nate Silver came up with. This shows that even a basic simulation can come marginally close to the results produced by more complex simulations probably trained on more data and information. Try this out with scores from other sports teams! As usual, find all of our code on GitHub. The SaberSmart Team
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