Super Bowl Hype Drive: Room for Squares

The run-up to the Super Bowl leads to a lot of great traditions -- parades! weird bloggers at media day! the disfiguration of millions of chickens! -- but high among these is the office squares pool. You've probably seen them; they look something like this:

You sign your name to a square, and once all the squares are filled, someone picks row and column values. Say you end up with the square (3, 1). That means that, at the end of any of the four quarters*, you win money if the score is X3-Y1 -- e.g., 21-13, 31-3, 41-33, etc.

* - Actually, the fourth quarter one usually includes overtime periods too, so if you had Ravens 8, Broncos 5 as a square in the AFC divisional playoff game, you would've won once Justin Tucker made that kick in the second overtime.

Obviously, the odds of every combination aren't equal, and you know intuitively that multiples of 7 or 3 are more likely to come up. Wouldn't it be nice to know what your odds of winning were compared to someone who drew, say, 7-7?

To determine the odds, I used Pro Football Reference's Score Index to find the score by quarters of all playoff games dating back to the 1994-5 season, when the NFL added the two-point conversion. This gives you a 20-year sample that (with this season included) encompasses 208 total games. Here are the total number of times each combination of numbers has occurred.

	0	1	2	3	4	5	6	7	8	9
0	62	16	6	66	65	6	25	112	7	7
1		5	3	13	20	6	7	27	10	3
2			0	3	4	1	4	9	3	1
3				19	35	3	11	64	6	8
4					20	8	9	43	9	11
5						1	1	7	3	1
6							1	14	2	3
7								41	9	6
8									2	3
9										1

Divide by the total number of quarters of football played (832), and you get percentages:

	0	1	2	3	4	5	6	7	8	9
0	7.5%	1.9%	0.7%	7.9%	7.8%	0.7%	3.0%	13.5%	0.8%	0.8%
1		0.6%	0.4%	1.6%	2.4%	0.7%	0.8%	3.2%	1.2%	0.4%
2			0%	0.4%	0.5%	0.1%	0.5%	1.1%	0.4%	0.1%
3				2.3%	4.2%	0.4%	1.3%	7.7%	0.7%	1.0%
4					2.4%	1.0%	1.1%	5.2%	1.1%	1.3%
5						0.1%	0.1%	0.8%	0.4%	0.1%
6							0.1%	1.7%	0.2%	0.4%
7								4.9%	1.1%	0.7%
8									0.2%	0.4%
9										0.1%

So, from the looks of things, (7, 0) is the best combination to get, right?

Not so fast. These numbers include ALL scores, both X0-Y7 and X7-Y0. In typical versions of this game, you only get one of the two squares. How can you determine which of those are more likely to occur? You can't really use home/away, since the playoffs include the neutral-site Super Bowl, and the home/away designation isn't meaningful for those games*. So let's just split it down the middle: if you have (7, 0) and someone else has (0, 7), 50% of the time you'll be on the right side of the pairing and 50% of the time you'll be on the wrong side.

* - Unless you really, REALLY like coin flips.

If you have (0, 0) -- or any of the other pairs along the diagonal -- you're in luck; there's no one else to split the odds with. That means that, overall, your best bet is that (0, 0) square, and THEN one of the (0, 7) or (7, 0) squares. The top five, by percentage:

Square	Pct
0,0	7.5%
0,7	6.7%
7,7	4.9%
0,3	4.0%
0,4	3.9%

The bottom five, of course, remains the same, with (2, 2) as the kiss of death with zero occurences.

Good luck, everyone. I for one will be rooting hard for a 42-12 final.