Earlier today I read someone comment about the Bills run of 4 straight Super Bowl losses, pondering how unlikely losing 4 straight title games really was.
This got me wondering, "What are the odds of winning a title for any playoff team?"
12 teams make the playoffs, if we treat every team as all being equal, then for the 8 wildcard teams the odds of a title maths-out as .5 chance to win their WC game * .5 for the divisional * .5 for the conference championship * .5 for the Super Bowl = 6.25%, 1 in 16. For the 4 teams with a bye the odds if all playoff teams are equal would be .5 for divisional * .5 for conf champ * .5 for Super Bowl = 12.5%, 1 in 8
So assuming all teams to be equal what are the odds a "hypothetical" team with 8 playoff trips, 3 of them with byes would win 1 title?
Well for each playoff trip without a bye there would be a 15/16 chance they didn't win, and a 7/8 chance they didn't win each time they had a bye. (15/16)^5 * (7/8)^3 = .485, a 48.5% chance they wouldn't win a title in that 8 year run.
Obviously all teams aren't equal, since the NFL went to a 12 team playoff in 1990 there have been only 5 Super Bowl winners who didn't have a 1st round bye. If the 8 wildcard teams actually had a 1/16 chance, we would expect them to have won half (10) of the titles, rather than just a quarter. This would suggest that the odds of a 3 through 6 seed winning a Super Bowl is really 1/32, with the bye teams reaping the benefits at 3/16 each, six times more likely to win a title.
With these tweaks are hypothetical team's non-title odds would look like (31/32)^5 * (13/16)^3 = .458. Still slightly more likely than not to win a title, but the margin's slim.
Homefield is worth something right? and better teams are likelier to have earned the home field advantage as well. In the 20 years of 12 team playoffs a #1 seed has won their conference 20 times. 1/2 odds compared to the 1/4 that would be expected if all playoff teams were equal. If we consider the Super Bowl a 50/50 shot since there's no homefield and seeding doesn't rank teams from different conferences against eachother so a #1 seed has much less meaning, then a #1 seed has a 1/4 shot at a title. This eats the #2 seed odds down to 1/8 from the 3/16 for all bye teams together.
Since the NFL went to 4 divisions in 2002, 2 teams have won their conference after losing their division. If all non-bye teams have a 1/32 chance of a title and we consider the Super Bowl a 50/50 shot a non-bye team should have a 1/16 chance to win their conference, exactly what we see from the wildcard teams (2 winners of 32 WCs). 3rd and 4th seeds don't seem to fair any better than wildcards when it comes to making the Super Bowl, and actually do worse winning it with both of those wildcards playing in the Super Bowl taking home the Lombardi. The '06 Colts are the only 3 or 4 seed to win the title in the last 8 years. It shouldn't really be a surprise that wildcards have at least matched the 3/4 seeds given that often one or both wildcards have a better record than a 4 seed from a weak division.
The recent history handicaps the Super Bowl as 1/4 odds for the #1 seeds, 1/8 odds for the #2 and everyone playing in the wildcard round at 1/32.
Our "hypothetical" franchise was the #1 seed in 2 of their 3 years with a bye so would be expected to miss out on a title over our 8 year span (2 #1 seeds, 1 #2 seeds and 5 other berths) at a rate of; (31/32)^5 * 7/8 * (3/4)^2 = .420, 42% of the time.
I imagine the identity of my "hypothetical" franchise is pretty transparent, but let's check out the odds of a 2nd title, since they were more likely than not to win 1.
For each of the 10 possible scenarios in which they won without a bye: (1/32)^2 * (31/32)^3 * 7/8 * (3/4)^2 =0.04%, by 10 scenarios is 0.4%
The 5 scenarios with non-bye win and a 2 seed win are: (1/32) * (31/32)^4 * 1/8 * (3/4)^2 = 0.19%, by 5 scenarios is 0.97%.
10 scenarios for a non-bye and a #1 seed win: (1/32) * (31/32)^4 * 7/8 * (1/4) * (3/4) =0.45%, but 10 scenarios is 4.5%. 2 scenarios for titles on a #1 and #2 seed (31/32)^5 * 1/8 * 1/4 * 3/4 = 2.0%, by 2 scenarios is 4.0% and winning both years they were the top seed (31/32)^5 * 7/8 * (1/4)^2 = 4.7%
For a grand total of... 14.6%, 1 in 7. How tough is winning 2 titles in 8 years? Even with a #1 seed every year the odds don't break 1/3.
Winning a title is hard, in the NFL, 31 teams fail every year, 15 of them playoff teams, 7 or 8 of those division winners, even among the teams watching from home on wildcard weekend, 3 or occasionally all 4 won't be lifting the Lombardi. Super Bowl or bust is catchy, but even for the best teams it's a set up for disappointment.