The ACME Freight Co. has containerized a large quantity of 4-gigabyte memory chips that are to be shipped to a personal computer manufacturer in California. The shipment contains 1,000 boxes of memory chips, with each box containing a dozen chips. The chip manufacturer calls and says that due to an error in manufacturing, each box contains exactly one defective chip. The defect can be detected through an easily administered nondestructive continuity test using an ohmmeter. The chip maker requests that ACME break open the container, find the defective chip in each box, discard them, and then reassemble the container for shipment. The testing of each chip requires \(1 \mathrm{~min}\) to accomplish.

a. Define a random variable representing the amount of testing time required to find the defective chip in a box of chips. What is the range of the random variable? What is the event space?

b. Define a PDF for the random variable you have defined in (a). Define a probability set function appropriate for assigning probabilities to events relating to testing time required to find the defective chip in a box of chips.

c. What is the probability that it will take longer than \(5 \mathrm{~min}\) to find the defective chip in a box of chips?

d. If ACME uses 28-hour-shift workers for one shift each to perform the testing, what is the probability that testing of all of the boxes in the container will be completed?