It’s Hip to Play Squares (Fri. 2/3/12)

Sunday evening over 100 million Americans will gather in living rooms and bars across the country to watch Super Bowl XLVI.  Some will root for Tom Brady, Rob Gronkowski (if he’s able to play), Bill Belichick, and the New England Patriots.  Some will root for Eli Manning, Victor “Salsa” Cruz, Tom Coughlin, and the New York Giants.  Many will be more interested in the Super Bowl commercials or Madonna’s halftime show.

 

Regardless of which team you’re cheering for, much of the most fervent rooting will have little to do with which team is actually winning the game at Lucas Oil Stadium in Indianapolis.  For many viewers participating in a Super Bowl squares contest organized by friends, co-workers, or the local watering hole provides more entertainment than the game itself.

 

There are many variations on Super Bowl squares contests, but they all start with a 10-by-10 grid featuring the numbers zero through nine along both the horizontal and vertical axes.  One axis is chosen to represent each of the teams playing, and in the simplest form of the squares game, participants pay a flat fee that buys them one randomly-assigned square on the grid.  At the end of each quarter and the end of the game the winning square is determined by the last digit in each team’s score at that point in the game. 

 

For example, suppose the contest organizer designates the horizontal axis to represent the Giants’ score and the vertical axis to represent the Patriots’ score.  Then, if the Patriots are leading 10-7 at the end of the first quarter, the person assigned to the (7,0) square would win a predetermined dollar amount or percentage of the total money raised by the selling of squares.

 

Even though the squares may be assigned randomly to participants, the history of the first forty-five Super Bowls has shown that not all squares on the grid are created equal.  Here’s a mapping of the 180 quarter-ending scores; I’ve randomly assigned the horizontal axis to the visiting team and the vertical axis to the home team in each of the previous games:

 

Home/Vis

0

1

2

3

4

5

6

7

8

9

0

13

2

3

11

4

3

4

10

0

3

1

2

0

0

1

3

2

1

3

0

1

2

0

0

0

0

0

0

0

1

1

1

3

6

0

0

6

4

0

5

8

0

0

4

7

1

1

2

1

0

2

7

0

1

5

1

2

0

0

0

0

0

0

0

1

6

2

0

0

2

0

0

2

2

0

0

7

10

2

2

5

3

0

4

7

0

3

8

3

1

0

0

0

0

3

0

0

1

9

2

0

0

0

1

0

1

0

0

0

 

Before converting this to a table of probabilities by square, it’s helpful to realize that assigning teams to the horizontal or vertical axis is an arbitrary exercise, so there’s no reason to believe that the probability of winning with square (0,3) should be any different than the probability of winning with square (3,0), even if that hasn’t been the case with the empirical data.  So, instead of using the 11 empirical “hits” with square (3,0) and the 6 empirical hits with square (0,3), it makes more sense to calculate square probabilities as though both squares had hit 8.5 times.

 

Once that adjustment is made, here’s what the table of square probabilities based on historical results looks like:

 

Home/Vis

0

1

2

3

4

5

6

7

8

9

0

7.2%

1.1%

0.8%

4.7%

3.1%

1.1%

1.7%

5.6%

0.8%

1.4%

1

1.1%

 

 

0.3%

1.1%

1.1%

0.3%

1.4%

0.3%

0.3%

2

0.8%

 

 

 

0.3%

 

 

0.8%

0.3%

0.3%

3

4.7%

0.3%

 

3.3%

1.7%

 

1.9%

3.6%

 

 

4

3.1%

1.1%

0.3%

1.7%

0.6%

 

0.6%

2.8%

 

0.6%

5

1.1%

1.1%

 

 

 

 

 

 

 

0.3%

6

1.7%

0.3%

 

1.9%

0.6%

 

1.1%

1.7%

0.8%

0.3%

7

5.6%

1.4%

0.8%

3.6%

2.8%

 

1.7%

3.9%

 

0.8%

8

0.8%

0.3%

0.3%

 

 

 

0.8%

 

 

0.3%

9

1.4%

0.3%

0.3%

 

0.6%

0.3%

0.3%

0.8%

0.3%

 

  

Note:  Blank squares have no “hits” during the first 45 Super Bowls.

 

 

Several observations:

 

  • The top 13 square probabilities account for over 50% of the winning squares (53.9%).

 

  • Almost half of the squares (46) have never been winners.

 

  • In reality the blank squares in the table above have a non-zero probability of being winners.  A larger sample-size would be needed to estimate these probabilities, but keep in mind that the NFL didn’t adopt the two-point conversion until 1994.

 

 

Enjoy the game!

The Sherpa

 

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