With the books closed on Super Bowl LI (51), and all of the parties over, I began thinking about the football grid pool. These pools are common in offices and at parties, and many people enjoy participating and eagerly await the random number assignments. In case you are not familiar with the pool, let me describe how it works. A 10x10 grid is drawn, and individuals can select or purchase (wink, wink) one or more of the squares. An example empty grid is seen in Graph 1, 10x10 Grid to Start.
Graph 1, 10x10 Grid to Start
When all 100 squares are claimed, the numbers 0 to 9 are randomly assigned to the rows and to the columns. The horizontal rows may be assigned to the AFC team, and the vertical columns assigned to the NFC team. Each cell will then represent a pair of numbers. If the square’s digits match the last digit of each team’s score at the end of a quarter or at the end of the game, then the owner of that square will be a winner.
Having participated (and lost) in many of these pools, I wondered what score combinations are better than others. I can make educated guesses knowing how points are scored in football, but I wanted to be more analytical. So, finding the scores from all of the past championship games, I created a distribution of last score digits by quarter for each NFL conference (see Graph 2, Individual Conferences).
Graph 2, Individual Conferences
What are the more likely combinations?
As most people would expect, scores ending with 1, 2, and 5 seem the most unlikely, especially early in the game. Scores ending in 0, 7, and 3 are much better squares to hold since field goals are worth 3 points, and touchdowns with the extra point are worth 7 points.
Looking at the 10x10 grid of the scores, we see combinations that have occurred. See Graph 3, Scores by Quarter to see a breakdown for all quarters. The combination of (0, 0) at the end of the first quarter is common, occurring in 14 out of 51 games. Remember, this could be a 0-0, or 10-0, 10-10 score, any score having the last digit be 0. The score ending in (2, 2) has never been witnessed at the end of any quarter or any game.
Graph 3, Scores by Quarter
How good or bad are your chances?
Now, what if you want to see what to expect with your squares in your grid pool? I used the Distribution platform of JMP Pro 13 to simulate 2,500 final scores for each conference. To do this, I fit a nominal distribution analysis for scores for each conference. Then, I right-clicked on the frequency counts and selected the Bootstrap option (available with JMP Pro). I then matched the AFC scores with the NFC scores to simulate the final score of 2,500 games and computed the probabilities for each combination of scores. Graph 4, Probabilities for Pairs of Scores shows the results.
Graph 4, Probabilities for Pairs of Scores
For the AFC, drawing a 4 seems to be the best number. For the NFC, 7 is the most likely digit. The combination of (4, 7) for the (AFC, NFC) scores is definitely the most likely combination of score digits. That combination occurs in more than 5% of the simulated combinations. If you have (5, 2) or (9, 2), I recommend you grab another chicken wing, watch the game and not look at the grid again. You have about a 0.15% chance of seeing these combinations!
My results can be reproduced with the attached data table, running the scripts included. And for more on this general topic, check out an earlier blog post by @XanGregg on visualizing Super Bowl scores.