An automobile assembly plant produces sheet-metal door panels. Each panel moves on an assembly line. As the panel passes a robot, a mechanical arm will perform spot welding at different locations. Each location has a magnetic dot painted where the weld is to be made. The robot is programmed to locate the magnetic dot and perform the weld. However, experience shows that on each trial the robot is only 85% successful at locating the dot. If it cannot locate the magnetic dot, it is programmed to try again. The robot will keep trying until it finds the dot (and does the weld) or the door panel passes out of the robot’s reach.

(a) What is the probability that the robot’s first success will be on attempts n = 1, 2, or 3?

(b) The assembly line moves so fast that the robot has a maximum of only three chances before the door panel is out of reach. What is the probability that the robot will be successful before the door panel is out of reach?

(c) Interpretation What is the probability that the robot will not be able to locate the correct spot within three tries? If 10,000 panels are made, what is the expected number of defectives? Comment on the meaning of this answer in the context of “forecasting failure” and the “limits of design.”