On 4th down, with 2 yards or fewer to go, New England has gained a first down on approximately 66% of its attempts with Tom Brady as quarterback. The Colts had one timeout. If the Patriots gain a first down, the game ends; they can slowly walk to burn a few seconds, then take a knee on each down to end the game. If they don't gain a first down, the Colts would still need to score a touchdown to win the game. Let's give the Colts a probability P of getting the six if the ball starts at the 28 yard line. So if the Patriots try for the first, their chance of losing is
(Probability of 4th down failure) x
(Probability of Colts scoring a TD from the 28 Yard line) = 0.33P
The average New England punt nets about 40 yards. Let's give the Colts a probability Q of scoring a TD on a driving starting at the Indianapolis 32. Then, the chance of the Patriots losing is simply Q. For Belichick's decision to make sense, we just have to believe that he gave his team a lower chance of losing. In math terms, that would mean 0.33P < Q. Doing some algebra leaves you with P < 3Q. In other words, for the Patriots to have made the right decision, we only have to believe the Colts odds of scoring a TD on a drive starting 28 yards from the end zone are less than three times the odds of the same outcome starting from 68 yards out. The win probability graph for the game suggests that, given 1st-and-10 from New England's 29, the Colts had roughly a 51% chance of winning in the actual situation. We have to believe that their chances under the punt scenario were above 17% for Belichick to have made a bad good decision. Considering the Colts' have scored touchdowns on 30% of their offensive possessions, my guess is that this was a good one.However, it felt wrong in the emotional/gut-instinct sense. Many times, we want to follow the rules, but when they conflict with our biases or convictions, we make an exception and hope to follow the rules next time.
Full Story: http://scienceblogs.com/cortex/2009/11/fourth_down.php