pr.probability - Football Squares

Dear Colleagues,

This is a math question for people who know the rules of (American) football.

Every year my barber runs a “football squares” game. He finds 100 customers, each put in 20 dollars, and each person is assigned a square on a 10 by 10 grid. After all squares are sold he picks numbers out of a hat to label each row and column 0 thru 9. So each contestant winds up being assigned with an ordered pair of numbers – in my case (3,7) this year. Now prizes are awarded by inspecting the last digit of the score of each team in the Super Bowl at the end of the first, second, third quarters and the final game score. Each winner gets $500.

For example, suppose that the scores are AFC 14, NFC 10 at the end of the first quarter. Then the person with (4,0) wins $500.

Now this is a fair bet, in the sense that each square is assigned randomly (and I believe that my barber doesn’t cheat.) However, it would seem that some numbers are “better” than others. Football scores are not random. For instance, at the end of the first quarter it is extremely unlikely that (5,5) will win. On the other hand, ((0,7) would seem like a good number. What is needed is some probabilistic analysis based upon the actual scoring patterns in football together with looking at actual scores of many pro football games. I am unable to find any analysis of this on the internet. I have tagged this as a probability question but since I work in operator algebras for a living this may be mis-tagged, and I ask you to pardon my error.

Let’s make it precise. Let f: {0,1,2,…, 9} x {0,1,2,…, 9} x {1,2,3,final} to [0,1] be the function that to a point (x,y,z) assigns the probability that the score at the end of the z’th quarter of the Super Bowl will be equal to (x,y) mod 10. Find the function f. Where does f achieve its max?

How about it, colleagues? Inquiring minds want to know!

CS