Chapter 4, Problem 4.191ACI

Optimal goal target in soccer. When attempting to score a goal in soccer, where should you aim your shot? Should you aim for a goalpost (as some soccer coaches teach), the middle of the goal, or some other target? To answer these questions, Chance (Fall 2009) utilized the normal probability distribution. Suppose the accuracy x of a professional soccer player's shots follows a normal distribution with a mean of 0 feet and a standard deviation of 3 feet. (For example, if the player hits his target, x = 0; if he misses his target 2 feet to the right, x = 2 ; and if he misses 1 foot to the left, x = - 1.) Now, a regulation soccer goal is 24 feet wide. Assume that a goalkeeper will stop (save) all shots within 9 feet of where he is standing; all other shots on goal will score. Consider a goalkeeper who stands in the middle of the goal.

1. a. If the player aims for the right goalpost, what is the probability that he will score?
2. b. If the player aims for the center of the goal, what is the probability that he will score?
3. c. If the player aims for halfway between the right goalpost and the outer limit of the goalkeeper’s reach, what is the probability that he will score?