Wildcard rematches – what are the odds?

This week’s NFL playoff schedule features a very peculiar scenario. Three of the four games are between teams matching up for the second time in seven days. Talk about your ultimate scouting report. So, what are the odds of that?

The odds 3 of 4 NFL wild card games will feature rematches from week 17 are one in 1,725.

Sounds pretty rare—but how rare? Well, it’s about as likely as being diagnosed with lung cancer in a year (1 in 1,668), about twice as likely as the odds of finding yourself on the waiting list for kidney donation (1 in 3,496), and about half as likely as winning ten bucks in the Mega Millions lottery (1 in 844, per ticket).

So, how’d we come to this figure?

Let’s start with each conference separately. Having rematches in 3 of 4 games means one conference will produce two rematches (this year it’s the NFC). So, let’s start with that one. The first thing to consider is which teams played in-conference games in week 17. There are typically 12 teams playing in conference during any given week. The probability of all 4 wild card teams playing conference games in week 17 is roughly 27%. Given that all 4 teams played in-conference games week 17, the probability of both games producing repeats is about 1%. So the probability that all 4 teams played in conference AND both wild card games produced a rematch is roughly (.27)*(.01) = 0.27%, or about one in 364.

The second conference is a little trickier, since only 1 game has to be a rematch, and thus more possibilities are involved. Here’s a simple breakdown of the chance of getting exactly 1 rematch AND having a specific number of teams that played in-conference during week 17:

  • 0 Teams and 1 rematch: 0%
  • 1 Team and 1 rematch: 0%
  • 2 Teams and 1 rematch: 0.66%
  • 3 Teams and 1 rematch: 4.4%
  • 4 Teams and 1 rematch: 5.5%

With no overlap, we can add these to get the total, which is about 10.5% or one in 9.5.

Since we now have the probabilities for each conference, we can multiply them together to find the final probability. Since we don’t care which conference has the 2 rematches, a factor of 2 is included to account for both possibilities: (.0027)*(.105)*2 = .00058. Therefore, the odds 3 of 4 wild card games will feature rematches of week 17 games are one in 1,725. That’s really unlikely (think “Cowboys winning a playoff game in the 2000’s” unlikely), and yet, that’s what happened this season*. If nothing else, this shows that there’s always a chance.

*Of course, these odds are for any random season for which all teams are considered equal. It also assumes the schedule is evenly distributed (taking divisional games into account would require many more calculations producing only a small change). In any particular season the odds will change week by week as you get closer to the finish and it’s easier to predict the teams that will take the wild card spots. This is one of those questions where there is not one right answer, but many, depending on your assumptions and methods. Our brain trust of mathematicians and NFL aficionados continues to debate how best to answer this question. This is our most reasonable estimate. What do you think?

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