Question: In the game of soccer, players are sometimes awarded a penalty kick. The player who kicks places the ball 12 yards from the 24-foot-wide goal and attempts to kick it past the goalie into the net. (The goalie is the only defender.) The question is where the player should aim. Make the following assumptions.
(1) The player's kick is off target from where he aims, left or right, by a normally distributed amount with mean 0 and some standard deviation.
(2) The goalie typically guesses left or right and dives in that direction at the moment the player kicks. If the goalie guesses wrong, he won't block the kick, but if he guesses correctly, he will be able to block a kick that would have gone into the net as long as the kick is within a distance d from the middle of the goal. The goalie is equally likely to guess left or right.
(3) The player never misses high, but he can miss to the right of the goal (if he aims to the right) or to the left (if he aims to the left). For reasonable values of the standard deviation and d, find the probability that the player makes a goal if he aims at a point t feet inside the goal. (By symmetry, you can assume he aims to the right, although the goalie doesn't know this.) What value of t seems to maximize the probability of making a goal?