The following example will illustrate the solution of a problem involving a normal probability.
The Radcliffe Chemical Company and Arsenal produces explosives for the U.S. Army. Because of the nature of its products, the company devotes strict attention to safety, which is also scrutinized by the federal government. Historical records show that the annual number of property damage and personal injury accidents is normally distributed, with a mean of 8.3 accidents and a standard deviation of 1.8 accidents.
What is the probability that the company will have fewer than 5 accidents next year? more than 10?
The government will fine the company $200,000 if the number of accidents exceeds 12 in a 1year period. What average annual fine can the company expect?
Step 1.  Set Up the Normal Distribution
 
Step 2.  Solve Part A From Table A.1 in Appendix A we see that Z = 1.83 corresponds to a probability of .4664; thus, From Table A.1 in Appendix A we see that Z = .94 corresponds to a probability of .3264; thus,
 
Step 3.  Solve Part B
Therefore, the company's expected annual fine is
