How does a team like the Steelers go from 0-4 to a dark horse for the final AFC playoff spot to inches away from clinching to the birth?  Then the Chargers who were a dark horse themselves go on an secure the 6th seed?

Philip Rivers mentioned, in an endorphin-high interview, that no one gave them a chance, saying the odds were against the Chargers.  This delves into the different realms of motivational and analytical that I want to stay away from, but how did these odds really work and how did they change through out the day?  Week 17 of the 2013 NFL season was a great example of Bayes’ theorem.

Here’s the math for Bayes’ theorem:

This is read: the probability of A given that B happens is equal to the probability of B given that A happens times the probability of A divided by the probability of B.  We are going to use this basic form of Bayes’ theorem to understand what happened to the Steelers and Chargers playoff odds throughout the day on December 29th.

The important point I’m going to illustrate here is that probability is not a property inherent in an event.  It is rather a guess or calculation based on known facts or frame of reference at the time.   When that frame of reference changes (in this case when the NFL schedule plays out) we can calculate new odds.  I’ll be using the above mathematical formula to calculate new probabilities as we discover new information through out the day for the Steelers.  Then I’ll compare this with a similar chart for the Chargers, who made the playoffs.

The first and largest problem for this exercise is determining the win/loss probability for the games.  I’ve searched online and there is much disagreement on what the playoffs odds where at the beginning of the day, let alone what any singular games odds were.  I could use Vegas odds as crowd-sourced odds, however, I’ll make this part easy and just make up numbers for illustrative purposes.  I have the Steelers and Chargers win probability weighted high because they were playing  the Browns and the Chiefs’ back-ups.  The Ravens and Dolphins I have at even odds of 0.5.  These could be endlessly debated, but let’s just assume they are correct.

Assumed probability chart for week 17.

For the Steelers to make the playoffs four things needed to happen.  First they had to win. [P(SteelersW) = .85]  Next the Ravens and Dolphins had to lose . [P(DolphinsL) = P(RavensL) = .5]  And then San Diego had to lose later in the day.  [P(ChargersL) = .75]  If you multiply all these together you will get roughly P(SteelersPlayoffs) = .05.  So there’s a 5% chance with the information at the beginning of the day that the Steelers will make the playoffs.

So let’s use Bayes’ theorem to calculate what the playoff are are if we know the Dolphins lose their game — P(SteelersPlayoffs | DolphinsL).

From just knowing the Dolphins losing their game, you can infer that the Steelers chances of gaining of a playoff birth is twice as likely as it was before.  I should also explain why P(DolphinsLoss | SteelersPlayoffs) is equal to 1.   This term assumes the Steelers made the playoff and asks what the probability is of the Dolphins’ loss given this information.  The Dolphins must lose if the Steelers make the playoffs so the term is equal to 1.  This will be true for every game we are considering.  (This problem becomes more complicated if there are multiple paths to playoffs, because the term will no longer be 1.)

Starting at P(SteelersPlayoffs) = .05, you can calculate the conditional probability in a chain as the Steelers’ win, Dolphins’ loss, and Ravens’ loss occurs.  Then the probability of the Steelers making the playoffs is calculated solely on the remaining game Chiefs/Chargers.  In-game win probabilities are calculated on advancednflstats.com.  They are dependent on time left in the game, score, and field position, and down, and they are independent of team skill.  This is the probability graph for the Chargers game.  I used this for the final two Steelers calculations: just before the Chiefs missed a FG, and then in OT during the Chief’s final drive after the Chargers kicked a FG to take the lead.

Steelers Playoff Probability — Area chart

You can see how the Steelers’ probability changed after each event, and how small the area was until the Chargers almost lost their game.  I will compare this with the Chargers, who had a better chance all through the day, except when the Chiefs were threatening to win.  The large green area is the probability at the end of the Chargers game when they won, and it’s value is 1.0 denoting they have clinched the playoff birth.

Chargers Playoff Probability — Area chart

To respond to Philip River’s on field comments about the Chargers having long odds, it’s misleading to think the Chargers somehow overcame those odds themselves, when they were the favorites to capture the final wild card spot when their game started.  It was the others teams’ losses that increased  the odds in the Chargers favor before they even played.

This exercise serves as an example of how and why probabilities change over time, and it illustrates how probability relies on known information or a reference point.  And how changing the reference point affects the known probability.