The NFL’s St. Louis Rams have announced they’ll hand over $100,000 to anyone who guesses the team’s exact schedule: whom the Rams will play each week and where the game will be played; whether it’s a Sunday, Thursday, or Monday game; and when St. Louis’ bye week will fall. Each NFL team plays 16 games in the regular season, and we already know the Rams’ 16 opponents—they’re predetermined based on the division St. Louis plays in, the team’s record from the previous season, and the NFL’s annual rotations of cross-division and cross-conference opponents. We also already know which opponents St. Louis will face at home, and which it will face on the road. Given those known parameters, this guessing game couldn’t possibly be as hard as Warren Buffett’s Billion Dollar Bracket Challenge, right?
Wrong. Picking the Rams’ schedule is much, much harder. If the Billion Dollar Bracket Challenge was impossible, then this is … really impossible.
If you’re guessing blindly, you have a (1/2)^63 chance of picking 63 NCAA tournament games correctly—that’s roughly 1 in 9.2 quintillion. Mathematician Jeff Bergen, though, estimates the odds of constructing a perfect bracket as closer to 1 in 128 billion, since you have a much-better-than-50-50 chance of picking a bunch of specific games (such as the ones between No. 1 and No. 16 seeds).
When picking a bracket, you can eliminate certain possibilities, such as a 16 beating a 1, based on probabilistic history. With the Rams’ schedule, every time you pick one game, your choices narrow. If you set St. Louis to play Seattle at home in Week 2, you know the Rams have seven home games remaining and that they will play the Seahawks just once more (in Seattle).
Using this methodology, I’ve put together a list of assumptions, and the resulting odds. Most of these assumptions are flexible. They are meant to narrow the odds to a reasonable ballpark (or football stadium). If you disagree with my assumptions, or think they can be refined, let me know in the comments and I’ll update the story with your thoughts.
Predicting division opponents
We know the Rams play each of their three division opponents (the Cardinals, 49ers, and Seahawks) twice each. We also know they will play a divisional rival in Week 17—the final game of the season. We’ll make the additional assumption that they won’t play the same opponent in back-to-back weeks.
Chance of predicting the Rams’ Week 17 opponent and location: 1/3*1/2 = 1/6
Chance of predicting the correct week for the other meeting between the Rams and their Week 17 opponent: 1/15
Chance of picking the two dates on which the Rams play their second division opponent: If we ignore the assumption that you can’t play a team in back-to-back weeks, the chance is 1/(15*14) = 1/210. The first meeting can be guessed correctly 1/15 of the time and—because one more game has been knocked off the schedule—the second game can be guessed correctly 1/14 of the time. If we do make our assumption of no back-to-back games, our chance (via Monte Carlo simulation) improves to approximately 1/105.
Chance of picking the two dates on which the Rams play their third division opponent: If we ignore the assumption that you can’t play a team in back-to-back weeks, the chance is 1/(13*12) = 1/156. If we do make our no-back-to-back assumption, our chance improves to approximately 1/100.
Predicting other opponents
The Rams have 10 other opponents and one bye week. The bye week occurs between Week 4 and Week 12. (For our purposes, we will assume all bye weeks are equally likely, though this has not necessarily been true in recent seasons.)
Chance of picking the bye week: 1/9
Chance of picking the weeks for the remaining 10 opponents: 1/10! = 1/3,628,800. Explanation: When there are 10 weeks and 10 opponents remaining, you can think of there being a 1/10 chance of picking each week correctly. If you get that right, it’s now 1/9, and after that 1/8, etc. When multiplied together, the result is 1/10!