# Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football

By Wayne Winston | Go to book overview

21
FOOTBALL DECISION- MAKING 101
During the course of a football game, coaches must make many crucial decisions, including the following:
 1 It is fourth and 4 on the other team's 30- yard line. Should we kick a field goal or go for a first down? 2 It is fourth and 4 on our own 30- yard line. Should we go for a first down or punt? 3 We gained 7 yards on first down from our own 30- yard line. The defense was offside. Should we accept the penalty? 4 On first and 10 from their own 30- yard line our opponent ran up the middle for no gain. They were offside. Should the defense accept the penalty? 5 What is the optimal run- pass mix on first down and 10?

Using the concepts of states and state values discussed in chapter 20, these decisions (and many others) are easy to make. Simply choose the decision that maximizes the expected number of points by which we win a game of infinite length. Let's analyze the five situations listed above.

1. It is fourth and 4 on the other team's 30- yard line. Should we kick a field goal or go for a first down?

To simplify matters we will assume that if we go for the first down we will get it with probability p (we assume that if we get first down we gain exactly 5 yards) or not get first down with probability 1 p. In this case we assume we gain exactly 2 yards. We define V(D [down], YTG [yards to go for a first down], YL [yard line where the team has the ball]) to be the number of points by which we will defeat a team of equal ability from the current point onward in a game of infinite length when we have the ball YL yards from our own goal line (YL = 20 is our 20 and YL = 80 is other team's 20) and it is

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