Math and playoff probabilities

I have not been asked what are the mathematical changes of the Bears and Packers playing for the NFC Championship though I would be asked that if they both win this weekend which they did.

In a 16 team conference the odds of any two teams meeting for a title would generally be close to 2/16 * 1/15 or 120-1 and it would be that simple if the NFL playoffs were like the NBA or NHL. However, the NFL playoff format changes things:

Note: All calculations are based on true math odds and not home field advantage or team history.

Four division champions from 4 team divisions make the playoffs plus 2 wild card teams which have the best record of non division winners. The Bears and Packers both make the playoffs if they both are wild card teams which hapens 1/2 * 2/12 * 1/11 or 1 out of 132 times or if one wins the division and one is a wild card team 1/2 * 1/6 (1 out of 12) or 1 out of 11 years combined which has happened under the current format in 2002 and also this year.

(Note: To make things less complicated than it already is, I am not going to figure in the unlikely scenario of both teams being wild card teams and advancing to the title game since the odds of that are 1/132 * 1/4 * 1/4 or 2,112 to 1. That changes the odds of both teams being in the playoffs to 12 to 1.)

The first round involves the two division champions with the worst records hosting the 2 wild card teams. About half the time (like this year) the NFC North champion does not have to play a wild cadr game so for the other team to advance the odds are1/2 (2 out of 4 possibilities) * 1/2 = 1/4. Then the chances are 1/2 that they don't play each other in the divisional round and 1/4 both winning so 1/4 * 1/2 * 1/4 = 1/32.

About one out of four times both teams play and both advancing is 1/4 (1 out of 4 possibilites) * 1/4 = 1/16 and then both teams winning again is 1/4 so 1/16 * 1/64 = 1/64.

One out of four times they play each other in the first round so one team has to lose. 1/4 (1 out of 4 possibilities) * 0 = 0

So the odds of both teams advancing to the NFC title game is about 1/32 + 1/64 or 3/64.

So the mathematical odds of the Bears and Packers (or any two divisional teams) meeting for the conference title is about 1/12 * 3/64 or 3/768 which is 256 to 1.