Math behind Belichick's 4th down gamble

The New England Patriots led 20-3 heading into the fourth quarter against the San Diego Chargers, and escaped with a 23-20 win after Kris Brown missed a 50-yard FG with less than a minute remaining that would have tied the game.

A key play in the game came with two minutes remaining when the Patriots, clinging to a 23-20 lead, faced a fourth and one at their 49-yard line. Bill Belichick decided to go for it, and BenJarvus Green-Ellis was stopped short of the first down marker, handing the Chargers the ball back with a chance to win or tie the game. The Patriots ended up with the victory, but did Belichick make the right decision in trying to convert on fourth down there?

If this situation sounds familiar, that’s because it’s the second straight season that Bill Belichick had the Patriots go for it on fourth down when leading late in the game with the ball in their own territory. Last year against the Colts on November 15, Belichick went for it on fourth and two in a similar situation, but the Patriots failed to convert and lost the game. Using a risk-reward analysis detailed here, we determined that Belichick actually made the right decision to go for it last year, but still lost.

We can perform the same analysis to see if Belichick was correct again. The decision of whether or not to go for it is based on comparing the Patriots chances of winning the game if they go for it versus their chances of winning if they simply decide to punt the ball.

Breaking down Bill Belichick's decision:

Using the win probability calculator on Brian Burke’s Advanced NFL Stats website, the Patriots had a 82 percent chance of winning the game by going for it on fourth down, based on the NFL and Patriots’ average conversion rates in that situation, which were very close to two-thirds in both cases. On the other hand, the calculator tells us that the Patriots had a 87 percent chance of winning the game if they had punted the ball.

So according to the analysis, this season Belichick actually did not make the statistically correct decision by going for it, but the Patriots ended up winning the game regardless. So it goes.