A couple weeks ago, Brian noted a scheduling fluke with the New England Patriots playing the first game of the team's divisional series at home over the last several years. Another was brought to my attention in Chris Brown's Fan Friday column from May 13 by a reader named Matt from Lancaster - the Buffalo Bills have faced nine of the last ten Super Bowl teams.
In 2010, the Bills narrowly missed an upset of the AFC Champion Pittsburgh Steelers and were destroyed by the Green Bay Packers. In 2009, the Bills knocked off the eventual Super Bowl runner-up Indianapolis Colts in a meaningless Week 17 game, while being pummeled by winner New Orleans Saints. In 2008, the Arizona Cardinals gave the Bills their first loss of the season. During the 2007 season, Buffalo lost to the Patriots and the New York Giants a combined three times. In 2006, both the Colts and the Chicago Bears beat Buffalo in their home stadiums.
Matt from Lancaster also asked Brown, "What are the odds of something like that happening?" Well, Matt, I have the answer.
Facing the AFC Champion
Through the divisional rotation the vast majority of the schedule is set before taking into account how bad a team was the previous year. The Bills have a 56.25% of facing the eventual AFC East champion every year playing nine of the sixteen AFC teams. There is always a shot that the Bills could be in the Super Bowl so we'll include them in the numbers even though they could never play themselves. Remember, only two of those opponents are determined by strength of schedule and the Bills haven't exactly been the cream of the AFC East crop in this time span.
Facing the NFC Champion
They have only a 25% chance of facing the eventual NFC champ and that is automatically assigned. They play four of the 16 NFC teams regardless of record.
Facing Both Conference Champions
In any given year that means the probability of facing both Conference champions is a hair above 14% (multiply the probability of one event, facing the AFC champ, by the other, facing the NFC champ). But this is only for one year.
Facing the NFC Champion In Five Straight Years
This is where we start to deal with some truly astronomical odds. Extrapolating the opposing conference number to five consecutive years, the odds of playing the opposing conference champion in five straight years is less than a .1% chance (.0009765625). Multiply the probability of the event happening one time (.25) by itself four times since it happened five times in five chances.
Facing Four Of Five AFC Champions
This gets a little technical because the Bills haven't gone five for five like they did with NFC Champs. The probability that an independent event will happen four but not five times is calculated using this formula:
n = number of trials (5 seasons)
r = number of specific events you wish to
obtain (4 AFC Champions)
p = probability that the event will occur (.5625)
q = probability that the event will not occur (.4375)
5C4 * (.5625)^4 * (.4375)^5-4
5C4 * (.1001129150390625) * (.4375)^1
5C4 * (.04379940032958984375)
[(5!) / (4!)] * (.04379940032958984375)
5 * (.04379940032958984375)
= .21899700164794921875 or roughly a 22% chance of facing the AFC Champ four of five years
Facing The NFC Champion In Five Straight Years And The AFC Champion In Four Of Those Years
The odds of any team facing their own conference champion four of five times and the opposing conference champion five times is calculated by multiplying the two probabilities together. The total comes out to be just two in ten-thousand teams will face that stretch. The probability rounds to .00021.
Man, that's bad luck. But the Bills share that distinction with the rest of the AFC East. Since all of these games were determined by the schedule and not the Bills' record, the same thing is true for each AFC East squad except the Patriots, who could not have played themselves in 2007 when they were AFC Champions. The Patriots did play the AFC Champion Steelers in 2008, though, where none of the rest of the AFC East did. Therefore each of the four AFC East teams have enjoyed this statistically improbable event over the last five years.
And virtually none of it was determined by the strength of schedule of the team's record. The numbers are based on all things being equal. The fact that the Bills continuously played the weaker AFC opponents would mean they should have a less likely chance of facing the eventual AFC champion that the Dolphins, Jets, and Patriots who have faced the better teams in the AFC over that stretch.
While I did all the math, I must acknowledge my brother Mark's help in this particular post. That Masters degree in math is finally paying off, buddy. He checked my math and made sure I followed the rules correctly.