Doing the math on Belichick's decision

Many are questioning Bill Belichick's decision to go for it on fourth down late in the fourth quarter of the Patriots' loss to the Colts on Sunday night, but is the decision defensible when looking at the numbers? Let's take at a stats-based risk-reward analysis of going for it in this situation.

There are four variables that need to be accounted for in making this decision:

1. The Patriots' chances of winning the game if they converted the fourth down.

2. The Patriots' chances of winning the game if they did not convert the fourth down.

3. The Patriots' chances of winning the game if they punted the ball.

4. The Patriots' chances of converting the fourth down.

The "Advanced NFL Stats" Web site was used to calculate the win probability. This metric uses NFL averages (taking into account time left, score differential, down and distance) and is not an exact representation of how likely the Patriots were to win, but does provide very good estimates for this purposes.

Let's go through it step by step:

1. If they converted the fourth down, the Patriots would have had a first-and-10 on roughly their own 30 with 2 minutes to go. Their average win probability in this situation would be 92 percent. You could argue that with the Colts only having one timeout left, the game would be over if New England converted, but there is a chance Indy would still get the ball back with a little bit of time left, so we'll stick with the 92 percent chance of winning the game.

2. If the Pats failed to convert the fourth down (which happened), they would give the Colts the ball (with a first-and-10) on roughly the New England 29-yard line with 2 minutes to go. The Patriots' win probability in this situation would be 66 percent.

3) If they punted the ball, using Patriots punter Chris Hanson's average of 44 net yards per punt in the game, the Colts would have gotten the ball (with a first-and-10) at the Indianapolis 28. The Patriots' win probability in this situation would be 79 percent.

4) The Patriots' chance of converting the fourth down varies based on how you look at it. The league average going for it on fourth-and-2 over the past two seasons (not including the Patriots' play Sunday) is 55.7 percent (49-for-88). The Patriots were 3-for-4 in that situation over the past two years entering Sunday night's final call. The Colts' defense allowed opponents two conversions in three opportunities on fourth-and-2 over the past two seasons entering Sunday night. Even though this indicates that the Patriots were more likely to convert than the league average, let's just say their chance of converting is the league-average rate of 55.7 percent.

Using these numbers, we can calculate the Patriots' expected win probability in both situations (going for it and not going for it):

Expected Win Probability When Going For It = (Probability of Success) * (Win Probability If Success) + (Probability of Failure) * (Win Probability if Failure) = .557 X .92 + (1-.557) * .66 = 0.805

Expected Win Probability When Punting (based on item No. 3 above) = 0.790

The result: .805 is greater than .790, so Expected Win Probability When Going For It is greater than Expected Win Probability When Punting.

Using these estimates, the decision is very close. The Patriots' expected win probability when going for it is greater than the expected win probability when punting, but by just 1.5 percent. If you say that the Colts and Peyton Manning had a greater chance of coming back after the missed fourth down conversion than the 34 percent given to them by the league average, this can be accounted for by reducing the .66 in the calculation above.

However, then you'd probably also give them better than the 21 percent chance to come back if the Patriots had punted, so you'd have to decrease the win probability when punting from 0.79. Most likely, you end up with a close decision either way. The "Advanced NFL Stats" Web site has its own (similar) analysis here.