Specifications for the shop welding of certain structural assemblies say that no welds with internal flaws greater than a critical volume can be permitted in order to guarantee adequate strength. An expensive x-ray inspection system is presently being used to inspect all welds. The test is considered to be absolutely accurate, and history shows that 15 percent of all welds are rejected and rewelded. The fabricator is considering introducing a new, less-expensive, but not perfectly reliable ultrasonic inspection device. By retesting known good and bad welds, the fabricator has found that the new device will, if a flaw is present, indicate this fact with probability 0.80 and erroneously pass the weld with probability 0.20. If no flaw is present, however, the device will pass the specimen with probability 0.90 and incorrectly indicate a critical flaw with probability 0.10.
(a) What is the probability of finding an indication of a bad weld using the sonic device? What are the probabilities of passing a poor weld and of a good weld being wrongly indicated as bad?
(b) If perfect inspection costs $1.00 per weld while the new method costs $0.50, and it costs $5.00 to redo a weld and $30.00 to have a defective weld leave the shop, should the new device be accepted?