There was much discussion about this on a local sports-talk radio station the other day. The host was convinced that only three teams are in Super-Bowl contention: the Philadelphia Eagles of the NFC and the Pittsburgh Steelers and New England Patriots of the AFC.
If all three teams were equal in talent and all other factors, then what is each team’s probability of winning the Super Bowl?
Intuition tells me that: three teams, each team has a 33% chance.
BUT, Pittsburgh and New England would have to compete against each other for a Super Bowl berth, while Philadelphia’s berth would be automatic. That means that Philadelphia has a one in two chance (50%). The other teams would thus each have 25% chance.
But that doesn’t seem right, either.
The host and his co-host argued quite extensively about this. The host insisted that his intuition was right: each team was one-in-three. The co-host insisted that his math was right.
This sounds vaguely like the Monty Hall third door puzzle.
Well, under these constraints, Philadelphia has a 100% chance of going to the Super Bowl. Since they’ll be evenly matched in that game, they have a 50% chance of winning.
The other two teams have equal chance of going to the Bowl (50% each). Once one of them gets there, that team has a 50% chance of winning. So each of the other two teams has a .5*.5=25% chance of winning.
Given the dubious assumption that any of these 3 teams beats any other team, and that their chances against each other are even, the correct conclusion is that New England and Pittsburgh each have a 25% chance, and Philadelphia has a 50% chance.
As for intuition: suppose there were 9 teams with an equal chance, but 8 of them have to duke it out for the right to face Philly in the Super Bowl. Does it make sense that Philly’s chance in that game is only 11%?
Probability doesn’t really work with this sort of thing. Perhaps Philadelphia is the only NFC team that looks like it’s going to the Superbowl. What is the probability they get there? 100%?
Well, what if Donovan McNab breaks his leg in the first play of the first playoff game? Are the odds the same? How about if he does it in the final game of the season? How about if he does it tomorrow?
Suppose Josh Parry tears up his knee at the same time?
So how exactly can you come up with a probability (other than giving odds – which is based on how people bet)?
And, of course, the playoffs are counted when you say “Any given Sunday.” We all know how well the invincible Baltimore Colts did in Superbowl III.
If you assume the teams will win all their games until they meet, you can come up with a number, but there have been plenty of times when a team had a stellar record at this point and didn’t make the Superbowl – and sometimes didn’t even make the playoffs.
Ultimately, there is no meaningful way to assign a probability to this.
Assuming the premises of the OP and barring the possibility of a tie (I have no idea if ties are possible, but I assume they are not – I’m obviously not an American football fan)…
Probability of Phili winning the Superbowl = 1/2 (winning the Superbowl / possible outcomes of the Superbowl)
For Pitt and NE, you have to calculate the odds based on the formula for a series of dependent events, which is odds-of-event-1 x odds-of-event-2, which in each case = 1/2 (winning playoff / possible outcomes of playoff) x 1/2 (winning Superbowl / possible outcomes of Superbowl) = 1/4
