Less Than A One in Hundred Chance – NFL, the Super Bowl, the Regular Season and Probability

There is less than a one in a hundred chance event that is taking place today. I thought I’d alert you to it as I try to restart my semi-frequent blogging exercise. The credit for the contents of this blog go to my son who told me about it, but thought it wasn’t worth writing up for others to see! So, what’s it all about?

The Seattle Seahawks are playing the Denver Broncos in a rematch of last year’s NFL Super Bowl (the ‘finals’ for American Football, for those who are wondering). The last time teams who met in the Super Bowl played each other in the next regular season games was in 1997! It has been sixteen years since that event has occurred.

Rare Super Bowl Rematch As Seahawks Host Broncos is the headline to an AP news story which appears in a number of different news outlets to capture the event. However, the headline notwithstanding, the article does not address the rarity of the event. Trey Wingo, of NFL Live tweeted –

How rare is a Super Bowl rematch the following season? Sunday’s @Broncos @Seahawks game will only be 6th such occurrence.— trey wingo (@wingoz) September 17, 2014

“…game will only be 6th such occurrence” doesn’t capture how rare it actually is. To understand how rare this even is we need a little NFL history, a little on the team structure of the NFL, how the NFL schedules games, a little NFL history and some very basic Statistics.

The National Football League (NFL) is divided into two conferences, The American Football Conference (AFC) and the National Football Conference (NFC). Each conference has 16 teams, slotted into four divisions (North, South, East, and West). Thus, the NFL has 32 teams. The Super Bowl is the championship game between the AFC and NFC champion. To become a champion you play 16 regular season games, and a few playoff games. Playoff games are only between teams in a conference (AFC teams play only AFC teams, and NFC teams play only NFC teams).

Regular season games can be inter-conference. In fact every team plays 4 inter-conference games every season. The way the games are scheduled, a team in one conference plays each of the teams from a division in the other conference on a 4 year rotating schedule. The way the games are scheduled every team in one conference will play every team in the other conference once in a regular season game every four years. This scheduling is structured and automatic. Thus there is a 25% chance that any team will play a team in the other conference in any given year.

The first Super Bowl was held in 1967, following the merger of the National Football League and the newer American Football League. Between 1967 and 1997, there was a rematch of the teams playing in the Super Bowl in the following season only 5 times. This hasn’t happened since 1997. In other words, it has taken 17 years for the teams that met in the Super Bowl to meet in the very next regular season.

So the question is, what is the probability that a game (or in probability terms an ‘event’) with a 25% chance of happening in a given year will not happen for 16 years and will occur in the 17th year? The answer is (0.75)16 * (0.25) = 0.0025, which is a quarter of a percent chance! So if you watch the Seattle-Denver game today please realize that you are watching something that is really rare.

A final note for inside the ballpark NFLers, and my colleagues who teach Statistics – in doing this calculation we have ignored the reorganization of the NFL conferences and divisions in 2002. We’ve assumed that 1997 to 2001 was organizationally the same as was post 2001. Taking that difference into account will change the result marginally, but the central story that this is really, really rare will stay on. My Stats colleagues who think it is worth making your students read this piece may want to ask them to go and research the organizational structure of the NFL pre-2002 and see how that will change this result!

P.S. The central ideas in this piece are Aditya Krishnan’s. So, I got a blog entry for the tuition payments we made!

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