Fun with numbers

A lot has been said around the NFL this year about the great season being had by the New England Patriots and the absolutely abominable season of the Miami Dolphins. Many wondered would we see the improbable: a team finishing 16-0 and another at 0-16 team in the same season. More reasonably, people have wondered what is more likely: whether the Patriots will win all sixteen or Miami will lose all sixteen. All interesting scenarios aside, what interests me most is how likely is it that a team could win or lose all of the games they play in one season. Some would have you believe that in competition as balanced as the NFL, it is sometimes luck that keeps a team from regression towards the mean. I can't say I disagree with that entirely.

If we assume that the Patriots are the best team in the NFL by a considerable margin (let's say the probability of them winning any single matchup is 95%, an unreasonably high number) then it still isn't very likely for them to proceed undefeated. With a 95% chance of winning each of sixteen games, this team still only has a 44% probability of winning all sixteen games. Likewise, if the Dolphins are that inferior to every team (or at least every team on the schedule) then they likewise have only a 44% chance of losing every game.

But consider that if a very good team that has a 3/4 chance of winning any single game (and randomness aside should have a very good 12-4 record) has only 1% chance of going undefeated (do the math, and that's assuming that this 12-win team isn't really a lucky 9-win team) and that should awaken us the awe of an undefeated season. If that doesn't amaze you, consider that when you bump the probability back up to 90% for each game, the likelihood of that team winning all 16 is under 20%.

And that's not taking into account home-field advantage and the equalizing effect of weather. It might be most reasonable to see a 16-game schedule as strata of opponents. I don't know how to divide teams as such but if the opponents were tiered into 4 categories of 4 each, with the Patriots' likelihood of winning at 95%, 90%, 85%, and 75%, the chance of winning out on that schedule is now less than 9%.

So as the Patriots push forward towards 16 wins we may do well to appreciate that the Patriots are quite good, as is any team that wins so many games, and also that if the Dolphins defeat them this weekend (which is a quite unlikely scenario) then maybe we should remember... that it isn't too unlikely.