01/31/2013 04:45 pm ET Updated Apr 02, 2013

A Probable Super Bowl

Millions of dollars will be spent to ensure that those watching the Super Bowl on Sunday keep their eyes on the screen. There is the game itself, of course -- there are sure to be multiple sideline reporters, dozens of camera angles, and scores of statistics in use to capture viewers' attention.

And no one will be turning away or muting the volume during the famous Super Bowl commercials. From Matthew Broderick's 2012 remake of Ferris Bueller's Day Off to the outlandish game of H-O-R-S-E played by Michael Jordan and Larry Bird in 1993, it is clear that Super Bowl ads are memorable and attention grabbing, indeed.

In spite of these efforts, there are still some things that will compete with the on-screen action on Sunday. Large spreads of snacks, interesting conversations at parties, and occasional calls of nature all have the potential to distract viewers from the game. What happens if an interruption causes you to miss a series of plays that led to one or more scores? Fortunately, you can use a little math to help you get caught up on the action!

Here is what we mean. Let's suppose you missed the entire first half of the recent AFC Championship Game between the New England Patriots and the Baltimore Ravens. When you join the game at the start of the third quarter, you see that the score is 13-7 with New England on top. You think to yourself, "I wonder how New England scored those points." There are a couple of natural first guesses:

  1. 7 + 3 + 3: a touchdown, an extra point, and two field goals
  2. 7 + 6: two touchdowns, one extra point, and one missed extra point
  3. There are some other possibilities as well. Among them are ...

  4. 8 + 3 + 2: a touchdown, a two-point conversion, a field goal, and a safety
  5. 7 + 2 + 2 + 2: a touchdown, an extra point, and three safeties
  6. 3 + 2 + 2 + 2 + 2 + 2: a field goal and five safeties
Your football intuition tells you that some of these scoring scenarios seem much more probable than others. The field of probability can help to quantify this intuition. To see this, let's consider the average number of times each of the following events occurs in an NFL game (where the average is computed using data gathered from the last three seasons).

Scoring EvenAverage Number Per Game
Extra Point (EP)4.84
2 Point Conversion (2PC)0.12
Failed EP or 2PC0.15
Field Goal3.31

Now, each team in an NFL game has, on average, 12 possessions per game -- a total of 24 possessions in all. The chart above indicates that there are approximately five touchdowns and three field goals in a game (on average). This means that roughly five out of 24 possessions (about one fifth) result in touchdowns and three out of 24 possessions (one eighth) result in field goals. We also note that since a safety occurs in roughly five out of every 100 games (or 2400 possessions), the probability of a safety occurring in a given game is 5/2400 (or 1/480). Similar reasoning shows that the probability of a two-point conversion occurring is about 12/2400 (or 1/200).

How can we use these probabilities to discern the likelihood of some of the scoring scenarios above? As is often helpful in mathematics, we first consider a simpler problem in order to get a better sense of the underlying situation.

Suppose we are playing a game in which we select gumballs, one at a time, from a bucket that contains five white, 10 red, and 15 blue gumballs. Once we choose a gumball, we note the color and then put it back in the bucket. If we were to do this five times, what is the probability that our five choices would have included two reds, two whites, and one blue?

Since 5 of the 30 gumballs in the bucket are white, each selection has a 5/30 (or 1/6) chance of being white. Similarly, each selection has a 1/3 chance of being red and a 1/2 chance of being blue. Since the selections are independent of one another, the probability that our five choices were, in order, red-red-white-white-blue is (1/3)*(1/3)*(1/6)*(1/6)*(1/2) = 1/648. There are other orderings (30 in all) that would produce the same color totals, and each such ordering yields the same probability (1/648). This means that the probability of choosing two reds, two whites, and one blue in our five choices is 30*(1/648), or about 4.6%.

Using similar reasoning, we can conclude that selecting three blue, two red, and no white gumballs has a probability of 13.9% -- roughly three times more likely than our earlier situation. This makes sense, since we see that the bucket contains more blue gumballs than any other color.

What does this mean for our football scores? What are the probabilities of the different scoring combinations that add up to 13 points? Just as we were 3 times more likely to select a blue gumball than a white one, a field goal is 60 times more likely than a safety and a touchdown is 96 times more likely than a safety. Using these probabilities, we find that the probability of getting 13 points with a touchdown, extra point, and two field goals is around 26%. The probability of getting 13 points with two touchdowns, one extra point (and one missed extra point) is 1.1%. This means that scenario (i) above is around 23 times more likely than scenario (ii). The striking difference here is due to the fact that missed extra points are rare.

Safeties are rare as well, and we can calculate scenario (i) to be about 424,000 times as likely as the touchdown and three safety scenario (iv). This certainly confirms our intuition.
What about other possible score totals? As you would imagine, as the totals increase, so do the numbers of possible ways to achieve the scores. For instance, there are 31 different ways to get to 27 points and there are 41 ways to get to 30 points.

How about the 1916 game between Georgia Tech and Cumberland? Georgia Tech scored 222 points to Cumberland's 0. There are 8,162 ways that a score of 222 can be obtained. Many of these scenarios are about as likely as the far-fetched shots made by Jordan and Bird in the commercial (for instance, 111 safeties) -- but they are possibilities nonetheless.

In the immortal words of Ferris Bueller, "Life moves pretty fast. If you don't stop and look around once in a while, you could miss it." Well, if you miss any part of Sunday's big game, don't just hit the rewind button. Use the opportunity to explore the possibilities with math! Even though your couch-sitting body isn't getting any exercise, you can give your brain a good mental workout!