A circus performer who gets shot from a cannon is supposed t

A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 185 feet and a standard deviation of 15 feet. His landing net is 50 feet long and the mid-point of the net is positioned 185 feet from the cannon. What is the probability that the performer will hit the net on a given night?

Solution

We first get the z score for the two values. As z = (x - u) / s, then as          
x1 = lower bound =    185 - 50/2 = 160      
x2 = upper bound =    185 + 50/2 = 210      
u = mean =    185      
          
s = standard deviation =    15      
          
Thus, the two z scores are          
          
z1 = lower z score = (x1 - u)/s =    -1.666666667      
z2 = upper z score = (x2 - u) / s =    1.666666667      
          
Using table/technology, the left tailed areas between these z scores is          
          
P(z < z1) =    0.047790352      
P(z < z2) =    0.952209648      
          
Thus, the area between them, by subtracting these areas, is          
          
P(z1 < z < z2) =    0.904419295   [ANSWER]