The ending of the game between the Indianapolis Colts and the New England Patriots (NFL teams) in Fall 2009 was quite controversial. With about two minutes left in the game, the Patriots were ahead 34 to 28 and had the ball on their own 28-yard line with fourth down and two yards to go. Their coach, Bill Belichick, decided to go for the first down rather than punt, contrary to conventional wisdom. They didn't make the first down, so that possession went to the Colts, who then scored a touchdown to win by a point. Belichick was harshly criticized by most of the media, but was his unorthodox decision really a bad one?
a. Use a decision tree to analyze the problem. You can make some simplifying decisions: (1) the game would essentially be over if the Patriots made a first down, and (2) at most one score would occur after a punt or a failed first down attempt. (Note that there are no monetary values. However, you can assume the Patriots receive $1 for a win and $0 for a loss, so that maximizing EMV is equivalent to maximizing the probability that the Patriots win.)
b. Show that the Patriots should go for the first down if p > 1 - q/r. Here, p is the probability the Patriots make the first down, q is the probability the Colts score a touchdown after a punt, and r is the probability the Colts score a touchdown after the Patriots fail to make a first down. What are your best guesses for these three probabilities? Based on them, was Belichick's decision a good one?