Math Quiz #2: Are You Ready for Some Football by Brad Boyle

Warning: This is a somewhat complex analysis of probabilities and statistical calculations. Enter at your own risk!


Well we have just finished our Superbowl contest won by yours truly (hold the applause please:)) with my prediction of a Pittsburgh Steelers victory and 45 total points scored in the game. I had based my total point guess on a final score of 31-14 and Pittsburgh actually won 27-23 resulting in 50 total points scored. However, with less than a minute to go in the game, Arizona was leading 23-20 and if they could stop Pittsburgh’s final offensive drive from scoring that would likely be the final score. I found this interesting because Maggie had predicated the Superbowl result would be 23-20 for Arizona! Wow. That led me to wonder, what are the odds that someone would correctly pick the winning team and exact score of Superbowl game? What do you think? I will give you a few guesses:

a) 1 in 100
b) 1 in 200
c) 1 in 1,000
d) 1 in 5,000
e) 1 in 10,000
f) 1 in 100,000

As a comparison, think how much easier it would be to predict the winning team and exact score of a soccer game. Then compare this to the chances of a perfect prediction for a hockey game or basketball game. How did you calculate the relative probabilities? Here are my thoughts.

While there is a huge number of technically feasible football game score results (such as 2-0, 3-0, 4-2, 98-2, etc.), the likelihood of some scores is obviously much higher than other scores. The best way to start to get an accurate probability estimate would be a statistical plot of all Superbowl results. Ideally if you are predicting the exact Superbowl score, you would like your statistical data to come from only Superbowl games where by definition you have two roughly similar talented football teams who have made the playoffs, won 2 or 3 playoff games and are under similar pressures. However, as there have only been 43 Superbowl games, you may wish to get additional statistical data to improve your analysis by including all NFL playoff games. You could increase your data set by including regular season games, but that introduces another variable where one team may be vastly superior to another team, producing scores that would not be statistically consistent with Superbowl scores.

Without having all this statistical data on hand though, I believe that the approximate odds of a perfect Superbowl prediction are about 1 in 1,000. I arrive at this conclusion with the following analysis:

I estimate that the highly probable range of total points scored by both teams in a Superbowl game is 25 – 55 points. That generates 30 roughly equal outcomes. For each of these total point outcomes there are roughly 40 ways that the total score can be achieved. For example, with a total score of 25 the game score could be 25-0, 23-2, 22-3, 21-4, 20-5, etc. There are 24 ways for this total to be achieved with each team having 12 winning scores. With a point total of 55, there are 54 ways for this total score to be achieved. On average, there are 39 ways for the 30 total point outcomes to be achieved{ (24+54)/2}. Therefore, the probability of correctly predicting the exact Superbowl score and winning team is about 1 in (39*30) or 1 in 1,170. This assumes that each of the ways of scoring the total points number are equally likely which we know is not true (this goes back to the point that a 55-0 or 53-2 score is less probable than a 28-27 final score) However, we have also excluded some low probability total scores which somewhat mitigates this methodological error term. I have rounded this down to 1 in 1,000 as my adjustment for this factor.

Based on this analysis, I estimate that Maggie was less than a minute away from a 1 in 1,000 Superbowl prediction. For anyone still reading this article, any thoughts or comments?