Superbowl Squares

As yet another service in my not a football blog, I’m going to talk about gambling a little bit, specifically the idea known as superbowl squares. What happens is, you have a ten by ten grid, “buy” a square within that grid for a certain amount of money (usually 5 or 10 dollars, however I’ve heard rumors of $100 per square or higher), then the each row and column has a digit between 0 and 9 randomly assigned to it, the rows represent one team and the columns another. Thus, each square has a value, for instance Patriots 2, Seahawks 9.

If, at the end of a quarter, the Patriots score ends in a 2 and the Seahawks score ends in a 9, that square wins, usually a quarter of the whole prize (so for $5 a square, the winner would get 125 (5×100/4).

So what numbers are the best ones? Lets assume several things.

1: Only touchdowns and field goals happen (ie, no safeties occur).

2: Touchdowns represent about 59% of all scores, field goals the rest.

3: Touchdowns are always worth 7 points (slightly unreasonable, touchdowns can be worth 6 or 8 points), and field goals are always worth 3 points (this one is true).

4: There are about 4 scores per team per game. (on average this is about right, but to truly do this you need not just the average information on the distribution of scores as well).

5: The score of one team has no effect on the score of the other team (again, this is false, a team trailing by 4 with 20 seconds to go, on a 4th and 20, is not going to kick a field goal, no matter what, a team trailing by 2 on a fourth down near the opponents end zone will absolutely kick a field goal).

6: The game can end in a tie (again, obviously false, and it should make all x-x numbers slightly less valuable)

The peculiar thing about football scores is that you can start a number line with 0, cycle through the numbers like this:

0, 3, 6, 9, 2, 5, 8, 1, 4, 7, 0

A field goal will move the active score one to the right, a touchdown one to the left. So a touchdown and a field goal will have no effect on the score. Then, all we need to do to get an estimate of the value of squares is as follows:

1: estimate the distribution of scores per quarter (I’ve set one up with a mode and average of 1 score per quarter, mostly normal distribution with a slight spike at 0 (to account for “negative” scores))

2: calculate the binomial distribution that within a given number of scores per quarter they will be touchdowns or field goals (so with 4 scores, 3 successes would be in excel =binomdist(4, 3, .59,false))

3: create a “tick” value for each scoring combination with field goals as +1 tick and touchdowns as -1 tick (so 4 scores, 1 touchdown would be a + 2 tick)

4: Multiply the probabilities in section 1 and section 2 together to get the probability of each scenario

5: Convert each “tick” value to its corresponding “point” value on the number line above, (so 0 becomes 0, 1 becomes 3, -2 becomes 4).

6: Total the probabilities associated with each point value

And you will get the probability of each end digit for the score for the end of each quarter. Multiplying them against each other gets the following grids:

End of First Quarter
0 7 3 4 6 1 9 8 5 2
0 21.05% 12.12% 8.37% 2.08% 0.99% 0.69% 0.23% 0.19% 0.09% 0.06%
7 12.12% 6.98% 4.82% 1.20% 0.57% 0.40% 0.14% 0.11% 0.05% 0.03%
3 8.37% 4.82% 3.33% 0.83% 0.39% 0.27% 0.09% 0.08% 0.04% 0.02%
4 2.08% 1.20% 0.83% 0.21% 0.10% 0.07% 0.02% 0.02% 0.01% 0.01%
6 0.99% 0.57% 0.39% 0.10% 0.05% 0.03% 0.01% 0.01% 0.00% 0.00%
1 0.69% 0.40% 0.27% 0.07% 0.03% 0.02% 0.01% 0.01% 0.00% 0.00%
9 0.23% 0.14% 0.09% 0.02% 0.01% 0.01% 0.00% 0.00% 0.00% 0.00%
8 0.19% 0.11% 0.08% 0.02% 0.01% 0.01% 0.00% 0.00% 0.00% 0.00%
5 0.09% 0.05% 0.04% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
2 0.06% 0.03% 0.02% 0.01% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%
End of Half
0 7 3 4 6 1 9 8 5 2
0 14.17% 8.09% 5.59% 4.12% 1.98% 1.96% 0.66% 0.64% 0.22% 0.21%
7 8.09% 4.62% 3.19% 2.36% 1.13% 1.12% 0.38% 0.36% 0.13% 0.12%
3 5.59% 3.19% 2.21% 1.63% 0.78% 0.77% 0.26% 0.25% 0.09% 0.08%
4 4.12% 2.36% 1.63% 1.20% 0.57% 0.57% 0.19% 0.19% 0.06% 0.06%
6 1.98% 1.13% 0.78% 0.57% 0.28% 0.27% 0.09% 0.09% 0.03% 0.03%
1 1.96% 1.12% 0.77% 0.57% 0.27% 0.27% 0.09% 0.09% 0.03% 0.03%
9 0.66% 0.38% 0.26% 0.19% 0.09% 0.09% 0.03% 0.03% 0.01% 0.01%
8 0.64% 0.36% 0.25% 0.19% 0.09% 0.09% 0.03% 0.03% 0.01% 0.01%
5 0.22% 0.13% 0.09% 0.06% 0.03% 0.03% 0.01% 0.01% 0.00% 0.00%
2 0.21% 0.12% 0.08% 0.06% 0.03% 0.03% 0.01% 0.01% 0.00% 0.00%
End of Third Quarter
0 7 4 3 1 6 8 9 5 2
0 9.92% 5.67% 4.62% 3.92% 2.36% 2.21% 1.16% 0.82% 0.43% 0.37%
7 5.67% 3.25% 2.64% 2.24% 1.35% 1.27% 0.66% 0.47% 0.25% 0.21%
4 4.62% 2.64% 2.15% 1.83% 1.10% 1.03% 0.54% 0.38% 0.20% 0.17%
3 3.92% 2.24% 1.83% 1.55% 0.94% 0.88% 0.46% 0.32% 0.17% 0.15%
1 2.36% 1.35% 1.10% 0.94% 0.56% 0.53% 0.28% 0.19% 0.10% 0.09%
6 2.21% 1.27% 1.03% 0.88% 0.53% 0.49% 0.26% 0.18% 0.10% 0.08%
8 1.16% 0.66% 0.54% 0.46% 0.28% 0.26% 0.14% 0.10% 0.05% 0.04%
9 0.82% 0.47% 0.38% 0.32% 0.19% 0.18% 0.10% 0.07% 0.04% 0.03%
5 0.43% 0.25% 0.20% 0.17% 0.10% 0.10% 0.05% 0.04% 0.02% 0.02%
2 0.37% 0.21% 0.17% 0.15% 0.09% 0.08% 0.04% 0.03% 0.02% 0.01%
End of Game
0 7 4 3 1 6 8 9 5 2
0 5.33% 4.31% 3.43% 2.98% 2.32% 1.65% 1.19% 0.82% 0.63% 0.43%
7 4.31% 3.48% 2.77% 2.41% 1.87% 1.33% 0.96% 0.66% 0.51% 0.35%
4 3.43% 2.77% 2.21% 1.92% 1.49% 1.06% 0.77% 0.53% 0.40% 0.28%
3 2.98% 2.41% 1.92% 1.67% 1.30% 0.92% 0.67% 0.46% 0.35% 0.24%
1 2.32% 1.87% 1.49% 1.30% 1.01% 0.72% 0.52% 0.36% 0.27% 0.19%
6 1.65% 1.33% 1.06% 0.92% 0.72% 0.51% 0.37% 0.25% 0.19% 0.13%
8 1.19% 0.96% 0.77% 0.67% 0.52% 0.37% 0.27% 0.18% 0.14% 0.10%
9 0.82% 0.66% 0.53% 0.46% 0.36% 0.25% 0.18% 0.13% 0.10% 0.07%
5 0.63% 0.51% 0.40% 0.35% 0.27% 0.19% 0.14% 0.10% 0.07% 0.05%
2 0.43% 0.35% 0.28% 0.24% 0.19% 0.13% 0.10% 0.07% 0.05% 0.03%

What can we learn?

The best numbers are 0, 7, 3 and 4, the worst are 5 and 2.

Numbers get more even the longer the game goes on, (this is born out by season ending statistics, which is essentially playing 64 quarters, as many teams have scores ending in 0 for the season (Giants with 380 and Arizona with 310) as have ending in 2, (Houston with 372 and Denver with 482)

Bad numbers are worse than good numbers are good. IE, 7-2 (one good number, one bad number), at the end of the game, is about as bad as 6-8 (two “ok” numbers). 0-2 (the best number and the worst number), is only marginally better than 1-9.

Even at the end of the game, only 33% of the numbers have positive expected payouts.

Pity to he who draws 2-2 (only .03% at the end of the game, which is possibly an overstatement, because the game cannot end in a tie).

Finally, I’d probably have done better to simply look at the scores at quarter end for each NFL game this year (or over multiple years) to get these probabilities, but I think my method has at least some value.

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