## An Almost Improbable Super Bowl

### February 6, 2012

For the second time in recent memory, the New England Patriots and the New York Giants delivered a competitive, exciting finale to the NFL season last night, from which the Giants emerged as Super Bowl champs. From a fan’s point of view, the game pretty much had it all: stellar individual and team performances, multiple lead changes, dramatic shifts in momentum, as well as some late-game tension and heroics (okay, maybe this isn’t from a Patriot fan’s point of view). However, upon further review, there was one aspect in which the game was, if not completely predictable, at least not as extraordinary as it might have been.

Let me preface my explanation by saying that I, like many of my fellow Americans, celebrated the evening with some casual small-stakes gambling. At the party I attended, a bunch of us agreed to buy into a Super Bowl Square pool. For those of you who haven’t seen this before, the rules vary depending on local preferences, but each square in a 10 x 10 grid usually corresponds to a pair of single digit numbers (e.g. 0-1, 9-8, 3-3, etc.). These number-pairs correspond to the final digit of the home and away team’s scores respectively. Each player “buys” a given number of squares for a given price and, if they “own” the square that corresponds to the final digits of the home and away team’s scores at the end of the game (or the half, or a quarter – *ymmv*), that player wins the pool (or some fraction thereof). Since the game is usually meant to be polite and accessible to party attendees who are neither serious about gambling nor football, the number-pairs are often randomly assigned after everybody has already bought their squares, but before the game has begun.

Now, if you stop to think about it, it should be pretty obvious that the second digits of football scores are not distributed randomly. Points can only be scored in combinations of 1, 2, 3, and 6, and there are some combinations which are historically more common than others. As a result, even though nothing has been decided in either the betting pool or the football game when the number-pairs are randomly assigned, some squares are now worth much less (are less likely to win) than others.

As it turns out, the 17-21 final score of the Patriots and Giants corresponded to one of the most likely combinations of second digits (7 and 1). I didn’t run the calculations myself, but pretty much every. single. person. who did agrees on the fact that 7 and 1 are, relatively speaking, much more likely than most of the 100 possible combinations of home and away team last digits. The precise extent to which people claim this combination is more likely depends on the method they use. I am not sure what the relative advantages and disadvantages of the various methods are, but among the sites I consulted, I intuitively prefer approaches like this one and this one, both of which were written by Dough Drinen and use actual game data since 1994 (when the current two-point conversion rules came into effect). If Doug’s done his math right, a 7-1 final digit outcome was among the ten most likely combinations. In the more precise language used in Doug’s analysis, the expected value of “owning” that square was quite a bit higher than the expected value of an average square.

So, why was the Patriots and Giants outcome *almost* improbable? When the Giants scored what wound up being the final touchdown with a little over a minute to go, the game seemed likely to finish with a relatively rare 7-2 combination once the Giants kicked the extra point. However (and here’s where both gambling and football get a bit more interesting), Giants Head Coach Tom Coughlin chose to have his team attempt a two-point conversion in an effort to reduce the possibility of a tie or a loss (should the Patriots somehow have scored a touchdown in the remaining minute). When the Patriots’ defense prevented the Giants from making the conversion, the score wound up 17-21 instead of 17-22, resulting in an outcome that was…well, almost predictable.

I find it hard to avoid thinking about the material impact of Coughlin’s decision: a truly awesome amount of money was riding on his choice when you aggregate all of the low- and high-stakes betting going on around a major sporting event like the Super Bowl. At a certain level, I guess that’s true at many points in the game, but it’s hard not to see it more clearly when the clock is ticking down and so much clearly depends on the outcome of a single play.

Of course, I have already said that the stakes were incredibly low in the specific case of my Super Bowl party. However, that did not prevent the lucky winner from appreciating the irony of the situation: the final digits in the score had suddenly gone from highly improbable to substantially more probable as a result of the failure of a somewhat improbable play that, on average, results in success a little more than half of the time (sorry if you need to read that last sentence twice). In case you haven’t already guessed it, that lucky winner was me!