The playoffs are a funny bit of business, where people tend to assume the #1 seed has a really good chance of making it to the Super Bowl. That is, unfortunately, not even close to the truth. If you ignore home field advantage, then it becomes easy to see that in these circumstances, the #1 and #2 seeds have 1 chance in 8 of winning (0.125), whereas seeds 3-6 have a 1 in 16 chance of winning (0.0625). But since in the playoffs, there is a home field advantage (at least until you reach the Super Bowl), the actual odds from Seeds 1 to 6 vary quite dramatically.
For now, we’re going to assume a home field advantage of 0.60. From 2001 to 2010, 100 non-Super Bowl playoff games were played, and the home team won 60 of them. This year, the home team won every time, unless the visitor was named the New York Giants, leading to a record of 8-2. So, I guess, the running total now, from 2001 to 2011, has to be 68/110, or 61.8% or so.
That said, I’m still going to use 60% in my calculations below.
For the sake of making it easier to turn any calculations into code, we’ll assign the home field advantage to the variable U (for “upper”), and to 1 – U, we will assign the variable L (for “lower”). Given these assignments, we now have:
LL = L*L
T23 = U*L + L*U
T45 = LL*U + (1. – LL)*L
Calculations of playoff odds
Seed 1 = U*U*0.50
Seed 2 = U*T23*0.50
Seed 3 = U*L*T23*0.50
Seed 4 = U*L*T45*0.50
Seed 5 = L*L*T45*0.50
Seed 6 = L*L*L*0.50
T23 is necessary to calculate the second game of Seed 2 or the third game of Seed 3. In this game, these two teams could face Seed 1, Seed 4, Seed 5, or Seed 6. Critically, they will either face Seed 1, for which they would be the visiting team, or all others, for which they would be the home team. The odds therefore become (odds of Seed 1 winning)(vistor’s odds) + (1 – odds of Seed 1 winning)(home team odds).
T45 is necessary to calculate the third game of Seed 4 or 5. In this game, these two teams could face Seed 1, Seed 2, Seed 3, or Seed 6. As Seed 6 is the only team for which Seeds 4 and 5 would be the home team, it is easiest to calculate the odds of Seed 6 making it to the third game, and then subtract those odds for the probability of playing as the visitors. Since the odds of Seed 6 arriving at game 3 are L*L, you end up with the formula given above.
Choosing a value of 0.60 for the home field advantage, we end up with:
Seed 1 : 0.18
Seed 2 : 0.144
Seed 3 : 0.0576
Seed 4 : 0.05184
Seed 5 : 0.03456
Seed 6 : 0.032
The range, from 18% to about 3%, is considerably more broad than the naive 1/8 to 1/16 values. Home field has a marked effect on the ability of teams to reach and win the Super Bowl. But the sheer number of teams involved, 12, and the arrangement of the playoffs, means that a #1 seed has, with a HFA of 60%, about a 36% change of making it to the Bowl, and a 18% chance of winning.
Note: this link has a coded version of the calculations above.